Pretty simple acheivement to obtain, load up Career mode and aim to win (I believe 1st, 2nd or 3rd is classed as a win) the first race in the Initiation Cup, I found these races to be a lot easier to win than the following championships.
To take a locked shortcut, you are going to need to find the key to unlock it, the key appears on every lap of the race, unless you are leading.
Belgium is pretty simple, make sure on your first lap you are in second place (or lower but another racer in front of you may grab the key beforehand so be warned) just before the waterslide/river bear left, follow the path round over the wooden bridge and collect the key (you will know you have it as it floats above your head). Follow the path to the start/finish line and after your first right turn, rather than go through the factory you should see a door to the right of the factory entrance, as long as you have the key above your head, aim for that door and it will let you through.
You completed 8 different championships in your career
Load up Career, race 8 different championships and come 1st, 2nd or 3rd by the time each championship has finished.
(Please note that to unlock some of the championships you must have earned Gold in a previous similar championship for example to unlock 'Mirrored 2' Championship, you must have earned a gold in the 'Mirrored 1' Championship.)
You completed 15 different championships in your career
Load up Career, race 15 different championships and come 1st, 2nd or 3rd by the time each championship has finished. (although this unlocked for me after my 14th championship!)
(Please note that to unlock some of the championships you must have earned Gold in a previous similar championship for example to unlock 'Mirrored 2' Championship, you must have earned a gold in the 'Mirrored 1' Championship.)
You completed 23 different championships in your career
Load up Career, race 23 different championships and come 1st, 2nd or 3rd by the time each championship has finished.
(Please note that to unlock some of the championships you must have earned Gold in a previous similar championship for example to unlock 'Mirrored 2' Championship, you must have earned a gold in the 'Mirrored 1' Championship.)
Load up Career, race 30 different championships and come 1st, 2nd or 3rd by the time each championship has finished.
(Please note that to unlock some of the championships you must have earned Gold in a previous similar championship for example to unlock 'Mirrored 2' Championship, you must have earned a gold in the 'Mirrored 1' Championship.)
Upon completing the 30 championships, as long as you have earned a bronze, silver or gold trophy in each, your career trophy cabinet will be full (you will see this on the screen before you choose which championship you want to race in the bottom right corner).
You completed a championship in your career in a splitscreen team
Grab a buddy with a controller (or family member, or just anyone with a controller!) and set up a 2 player career, choose a championship and come top 3 by the end of it.
I did this with teams switched on and came 3rd in a championship.
You used power-ups against 3 drivers who are within range of the safety car (globals not included)
Some people have difficulty with this achievement, the simplest way is to stay up with the pack at the front or if your leading slow down to join the pack, if your lucky enough to have a Pulse power-up, use it in proximatey of 3 other racers.
This can be a very easy achievement if you have 2 controllers and can be done from the Play option of the game and doesnt have to be done on a set CC class.
Choose a race (Belgium is a good choice) and press to customise the race, change the Power-up Balance to Bubbles only and accept.
When picking your driver, choose Lewis Hamilton or Jenson Button as they can both swap their power-ups, the second player can choose any driver.
Whislt in the lobby change the setup so its set to teams have yourself on one team and player 2 as the other team, then start the race.
When the race starts, allow the CPU drivers to make there way off the line with player one following, player 1 grabs a power up and hopefully grabs a 3x Ricochet power up, if you dont get it, hold to change the power up, if your unlucky again driver to the next power up and so on until you get 3x Ricochet Bubbles
Try to parkthe 2nd player infront of the first, then with player 1, fire the Ricochet Bubble 3 times (pause before firing each one so that they each register).
When you notice your car leaving the floor after a jump release , then just before the car lands hold , you should receive a small boost (Known as a jump boost) you will know if you have done it right due to the small boost followed by a badge popping up on the left side of the screen.
Slipstreaming is where you stay behind a driver, you will gain speed (and if you are Seb Vettel or Mark Webber a boost power up) once you have gained enough spped, overtake the racer you are slipstreaming to obtain this little beauty.
You can obtain this through the 'Play' mode. (Credit to loserb0i)
38 Shortcuts to take
Belgium
2 Shortcuts
1. Just after the factory take an immediate right (most AI cars go this route by default it seems).
2. Get the key by following the path which starts immediately to the left of the rapids, and use the key to the right of the factory entrance.
Germany
4 Shortcuts
1A. At the start of the race there is a dirt road off to the left by the first corner, if you time it right you'll get the shield key.
2. After you drop back down from getting the key, there is a shortcut immediately to the left, keep you're eye out as it is quite easy to miss.
1B. Take the furthest right path through the castle for the locked shortcut.
3. There is a big jump that you can land on a high route up through the top of a castle, you need a full KERs corner boost to make it.
4. Just before the final couple of corners there is a small narrow road to the right which cuts them both (behind a little blue hut).
Italy
3 Shortcuts
1. Right at the beginning go to the left to skip the first chicane.
2A. Collect the key, as you enter the Colosseum take the path on the right.
3. After collecting the key to the locked shortcut, the road splits, to the right is a group of charging shields, to the left is a little ramp, take the ramp.
2B. After the long straight where you weave between the broken bits of the bridge, they'll be a left hand corner, stay on the right and keep your eyes open for a gate.
Brazil
4 Shortcuts
1. At the first corner you want to cut across the grass and hit the boost pads.
2. After the next KERs corner take a sharp right, jump up on the roof to the right and get the key. in the big round area of the carnival there is what looks like a chicken head towards the back left, that's the gate to go through.
3. On the big wooden part of the track there is a narrower path off to the left that goes across the pond.
4. 2nd lap just before the carnival take the ramp on the left to jump onto the roof and stay on it all the way to the end.
Abu Dhabi
4 Shortcuts
1. Just after the roller coaster there is a chicane, head towards the right and go over the gravel trap to skip it.
2. Collect the key from just inside the first building, there is a path immediately to the left as you enter, use the key a little bit later before exiting the building on the left.
3. On the next lap just after you come off of the moving platforms held up by cranes, you'll have to go around hairpins in the sand, after the first there is a path in the wall on the right.
4. Immediately after the last shortcut there is a KERs corner, charge your boost and go straight forward, aim for the ramp, and jump over the puddle.
Monaco
3 Shortcuts
1. Just after the fountain area take a sharp left (just before there is two moving vehicles), you'll go through a shopping mall kind of area, this is where you collect the key, as you come back onto the road take a right through the gate (just before the next hairpin).
2. Immediately after the locked shortcut take a right up onto the path (you'll go over a swimming pool).
3. On lap 2 after the tunnel there is a chicane, cut across the grass on the right to skip it.
Singapore
1 Shortcut
1. In the fountain area take a right for an alternative route, this will get you the key. just after the start/finish line there is a split section where the track is divided by a wall as you go down stairs, take the left path and go through the gate.
Australia
3 Shortcuts
1. Just after the stadium at the beginning take a right out onto the docks and ride across the piers.
2. On the straight with trams take a left path to find the hidden key, use it as you enter the sharks mouth on the right hand side.
3. On the next lap back on the straight with the trams there is a jump over a sculpture on the right.
USA
9 Shortcuts
1. As you get to the dirt track area, head to the left where there is a ramp.
2. When you see the first set of tractors, head right through a barn.
3A. Get the key from the jump to the right of the first pit lane.
4. After the dam cut the first 90 degree corner.
3B. Take the locked shortcut, its a gate just to the left of the last shortcut.
5. Head straight through truckasaurus' legs and ride over the slow dirt behind him.
6. 2nd lap, during the canyon area before the dam, ride the higher path on the left all the way to the dam.
7 & 8. Towards the end of the lap (just before truckasaurus, you slalom around a couple of field goals, cut across two pieces of grass here for two shortcuts.
9. 3rd lap, in the canyon area, they'll be an opening to cross a small stream, cross it.
Britain
3 Shortcuts
1. Near the start, the road forks, to the left tarmac, to the right it drops down onto dirt and has 3 red boost pads, take the dirt road.
2A. Collect the key from inside the first factory, there is a path immediately to the left upon entering.
3. After jumping out of the factory you slalom around 3 hay bales, planes fly by and blast the hay bales away, you want to time it so the planes have blasted the hay away before you get there, and cut across all 3 corners
2B. Take the shortcut, as you enter the first airfield, at the back right of the field is a big plane you cut through.
Japan
2 Shortcuts
1. In the area with the robot girls on wheels go out of the building to the right, follow the path to get the key. use the key just after the start/finish line through a big red semi circle fan.
2. On the 2nd lap, right after the building with the Japanese robot geishas just outside of that is a grass shortcut to take.
This achievement does not need to be done in Career Mode, it can be obtained via Play Mode.
The simplest way I found was to set up a race, whilst choosing the track, press to customise the race, change the race to a Slalom and Point Target to 10,000.
Enter the race and aim to hit the same colour gate each time you see one, the more you hit the higher the points (max of 500 per gate) someone told me that waiting for the other racers to move ahead would make it easier to hit the gates, however this didnt work for me as the driver at the front would aim for the Star Gate gaining 500 points every gate and beating me to the 10,000 target, I found it easier to stay with the pack and either hit the star gate or aim for the same colour, it took me about an hour to get all tracks done this way.
Just make sure that before you customise the race you are pressing on the track you need to achievement for!
Aim to get a Gold Trophy on all Time Trial tracks, for more help on this please see the below video, its really useful as most Golds are not easy to obtain as you will more than likely be a second or 2 out!!
You earned 1,000 Exhibition points in under 5 seconds
The best way to earn 1000 points in under 5 seconds is to make your way to the front of the pack, the higher in the leaderboard you are the more points you get, you will also get points for staying in first place and also overtaking.
Lap Amount - 3 (you can choose to have more if you wish)
Power Ups - None
Damage – Disabled
Modifiers - None
CC – 1,000
When the race starts avoid ALL fuel pickups until you reach the two fuel boosts on the outside of the corner just before the second pit-lane. This one will see you through the finish line and will see your tank empty about 10ft before the line.
Aim to get all KERS boosts and try to get a good lead from the start, all jump boosts are also essential!
This can be done in 'Play' Mode with a customized race.
Select Belgium (in my opinion the easiest track) and press to customise the race.
Lap Amount - 9 (you can choose to have less if you wish)
Power Ups - Normal
Damage – Disabled
Modifiers - 'Project Orion' and click to add it (Permanent Boost)
CC – 1,000
The achievement pops as soon as the whole track is owned, therefore you do not need to finish the race should you get it before then.
Belgium has 7 sectors to grab, the only one that might catch you out is the one through the locked shortcut, so be sure to pick up the key situated on a path to the left as the rapids start, and use it to the right of the factory entrance.
Singapore also seems to be a popular choice to get this achievement however not tested this myself (Credit to Blue n Gold Eel).
Im not sure if this can be obtained on your own as I obtained it during a 2 player race.
Eitherway in a 2 player race, stay towards the back of the pack, once the Eliminator timer starts and you get eliminated you become a Ghost Racer with an infinite supply of Pulse Power Ups. All you need to do it hit another live racer with the pulse before they too become a Ghost.
In 2 player mode ensure that one of you is leading the pack as once you both become Ghosts the game ends (hence why I dont think you can get it in a Single Play session).
In Trophy Chase, the aim is not to win the race but obtain as many Trophies (scattered on the track) as possible, if you receive any damage you will drop trophies.
Aim for the racer with the most trophies and hit them with a power up, Pulse power-ups work the best due to the range around them and as you will all be jostling for trophies, hitting someone with it shouldnt be too difficult.
For this achievement you are required to win 6 races in a row in local multiplayer.
Firstly plug in a second controller
Choose the 'Play' mode from the main menu
Select Belgium (in my opinion the easiest track) and press to customise the race.
Lap Amount - 3 (you can choose to have more if you wish)
Power Ups - None
Damage – Disabled
Modifiers - 'Project Orion' and click to add it (Permanent Boost)
CC – 1,000
The only tip I can give you is to not to take the shortcut after the Factory, take the proper route and use the KERS boost to your full advantage (with Project Orion this will get you max boost!)
Remember also that you can hit restart whilst going for this achievement and your win streak will not reset. (Credit to Blue n Gold Eel)
Once you have been hit by a power-up, you will start to notice that your car has damage and isnt going as fast as it was before the power-up, luckily for us, there are pit stops (yes stops, meaning more than one per lap!) These are indicated by a spanner and big yellow arrow directing you to the pit lane (you will also get notified when you should go to the pits).
You used a power-up against every opponent on the track (globals not included)
As the title reads, aim to hit each racer with a power-up, this can be done in the 'Play' Mode, I would suggest setting it to 10 laps or so and having a list of all racers taking part, then once you have hit one of the racers pause your game and mark them off.
When racing, hit a power up and hopefully you will get a Richochet Bubble or 3x Richochet Bubble (Yellow Bubble) I got this completey accidentally, I used the power up and somehow managed to get hit by it on the next lap!
I have been assured you can get this by driving up to a barrier and firing it straight (so it should bounce back and hit you) however I have not tested this way of doing it.
During your play through of F1 Race Stars you will obtain badges for things such as hitting racers with power-ups, boost Starting, Fully Completing KERS boosts etc.
These do stack up over time, so the more time you put into game modes the sooner you will get this achievement (I reckon you could easily have this by the time you complete Career Mode).
(1) 
to customise the race, change the Power-up Balance to Bubbles only and accept.
to change the power up, if your unlucky again driver to the next power up and so on until you get 3x Ricochet Bubbles
, then just before the car lands hold 