Refer to the costume guide by MightyMango here

There are 79 costumes in total to unlock. Most costumes will be unlocked by the time you finish the Shhhh It’s a Secret achievement. The rest can be unlocked by getting 100% perfects all songs on one of the three difficulties. (That is full combo, all perfects. You can check this by looking at the song stats after the song is finished. Accuracy should be 100% and Max Combo and Perfects should read the same number.)

After every few songs completed on any difficulty, a character will unlock. Unlock all the characters for this achievement.

Go to Create a Wildcat and create a new male character. Just change some settings and save. It should unlock right away.

Same as Homecoming King, except create a female character.

Complete all songs on Sophomore and Junior difficulty.

Complete all songs with a score of A+ on Sophomore and Junior difficulty. To get an A+ you need a minimum of 97% Great/Perfects.

Senior difficulty unlocks when you complete all songs on Junior difficulty.

Just go through all the quizzes. It doesn’t matter what results you get. I suggest just pressing A repeatedly to get the left-most conclusion. This will save you time for the Well-Rounded Wildcat achievement.

Go to tutorial menu and finish all of the tutorials. 

Every song has a secret cue to be found by activating the superstar mode at the right time. The cues will change to Wildcat logo instead of the normal green or pink circles. There are 29 songs and 3 difficulties, so you need to find 81 secret cues. This achievement unlocks when you found all secret cues for the Junior and Senior difficulties, but to get the Shop-a-Holic achievement you need to find all Sophomore secret cues as well.

See Well-Rounded Wildcat.

Signatures unlock when you play using the character a lot.  I do not know the exact requirement unfortunately. Just keep playing with the characters that you do not have signatures for. If the achievement does not unlock for you after getting the 8th signature. Go out to the dashboard, and come back. Play one song and it should unlock at that point.

Finish Junior difficulty and get the required number of credits. Should unlock for you at the end of Junior year if you played through Sophomore first or during Senior year if you started with Junior difficulty.

If you play well, you can activate Superstar power by pressing + when the Star on the upper left is full. Do this until you have spent two hours in this mode. You can check how you are doing by going to the yearbook / student records. It’s on the second page.

Hold cues are the ones where you have to hold down a single direction for a few seconds. You have to spend one hour total in hold cues. You can check how you are doing by going to the yearbook / student records. It’s on the second page.

Just hit 1,000 cues total. You can check how you are doing by going to the yearbook / student records. It’s on the second page.

Hit 1,000 cues total with Perfect grade. You can check how you are doing by going to the yearbook / student records. It’s on the second page.

If you play well, you can active Superstar power by pressing + when the Star on the upper left is full. Do this 100 times total. You can check how you are doing by going to the yearbook / student records. It’s on the second page. 

Get a score of A on every HSM1 song. Sophomore and Junior difficulties need to be cleared.

HSM1 songs are:
Bop to the Top
We’re All In This Together
What I’ve Been Looking For (Reprise)
What I’ve Been Looking For
Breaking Free
Get’cha Head in the Game
When There Was Me and You
Start of Something New
Stick to the Status Quo

Get a score of A on every HSM2 song. Sophomore and Junior difficulties need to be cleared.

HSM2 songs are:
What Time Is It
Work This Out
I Gotta Go My Own Way
You Are the Music In Me
Everyday
I Don’t Dance
Bet On It
You Are the Music In Me (Reprise)
All For One
Fabulous

Get a score of A on every HSM3 song. Sophomore and Junior difficulties need to be cleared.

HSM3 songs are:
Now or Never
I Want It All
Right Here Right Now
Can I Have This Dance
The Boys Are Back
Just Wanna Be With You
A Night to Remember
Scream
Walk Away
High School Musical

Cumulative total of 100 million points in any song/difficulty combined. You can check your progress by going to Yearbook / Student Records.

Total game time of eight hours - Only the time actually playing songs counts. You need to actually play ten hours for one of the costumes.

There are three conclusions to every quiz. Left can be gotten by picking left all the way through. Right can be gotten by picking right all the way through. Middle can be gotten by picking right until the last question, pick the left one for that. Do this for all of the quizzes to get the achievement.

Get an A+ on one song in Junior or Senior difficulty. You need to hit 97% of the notes to get this score.

Get a full combo on one song in Junior or Senior difficulty. You need to get all Perfects and Greats.

50% of the notes must be Perfect on one song in Junior or Senior difficulty. If there are 100 notes in the song, you must have 50 Perfects for this to unlock. The number of perfect notes can be seen on the after the song stats screen.

Choose What I’ve Been Looking For (Reprise) on Junior difficulty and hit the down cues only.

Choose any song on Junior or Senior difficulty and put down the controller. 

Choose any song on Junior or Senior difficulty and pass it without hitting hold cues. Probably easiest to do What I’ve Been Looking For (Reprise) as this song is very short.

Every song has a secret cue to be found by activating the superstar mode at the right time. The cues will change to Wildcat logo instead of the normal green or pink circles. Do this on a song for Junior or Senior difficulty. This cannot be done on Sophomore.

See 300 Times the Dancer.

See 300 Times the Dancer.

Continuing to hit the notes without "Oh Nos" will get the combo counter in the bottom right going. Go all the way up to 300 in a row to get this achievement. There are only a few songs in the game with more than 300 notes. All three of these achievements must be done in Junior or Senior difficulty. The songs with more than 300 notes are:

All For One (Senior)
High School Musical (Senior)
I Want It All (Senior)
Stick To the Status Quo (Senior)
The Boys Are Back (Senior)

If you play well, you can active Superstar power by pressing + when the Star on the upper left is full. Do this three times in one song on Junior or Senior difficulty. This is not achievable on some of the shorter songs. I suggest a long song like High School Musical or The Boys Are Back.

Choose a song you are comfortable with on Junior or Senior difficulty. Intentionally miss notes until the meter is in the red zone and alarm will sound warning you that you are almost failing. Start playing from here to max out the performance meter all the way to the right until the end of the song.

Choose a song you are comfortable with on Junior or Senior difficulty. Make sure the performance meter doesn’t fall below the yellow zone.

Play a two-player game of Dance Duo in which that you do twice as well as the second player. If you are playing alone, this should be really easy.

Play a two-player game of Dance Duo in which that you get a 100 note combo. Combo meter is separately counted so you only need to play with one controller.

Play a two-player game of Dance Off. In Dance Off, instead of Superstar you get Blocker. Press + to block your opponent and get this achievement.