Here's some tips that will hopefully help you with the quest mode. 

You should be playing quest mode in at least the difficult level, and you should be trying to prepare yourself to move up to expert level. The reason for this is you need to score enough points in each stage to pass it, and on the beginner and basic levels it's literally impossible to pass boards even if you combo the whole song. This said, on the difficult level the highest scoring song I found was Hey! On the expert level it's also the highest scoring song I found, but Rockit, Walk Like an Egyptian, and I Know You Have Soul, are high scoring songs that are good songs. I didn't check all 70 songs because I don't like a lot of them, but if you find any higher scoring songs let me know and I'll edit this guide.

You should spend all the money you win on obtaining the songs. Go to the shop and buy them for 2000 each. This increases your score on the song you buy by about 50%! That's huge. In the beginning only buy songs you are going to use. If you finish buying all the songs, start buying the videos for 500 each. If you put the game on the easiest setting like I suggested in the first post, you won't need these videos for extra points though.

You can save a ton of time by exiting a board as soon as you complete the goal. You will not be penalized at all. Simply hold the back button down, and you will go the "board cleared" screen.

Very important- they don't explain this in the instruction book, and I know I'm saving someone time here: When you're choosing a song for a dance battle, above the name of the song there is a blue note if you have bought the song (and it gives you a large score advantage like I explained above). But if your opponent has the song there is a red note above the song. It is possible for both you and your opponent to have the song. You want to pick a song that you own, but your opponent does not. During the song, you will see if you own the song because the dance meter will be striped orange and yellow- if your opponent has the song, his part of the meter will be striped white and blue.

Check the difficulty setting before you do a "special offer." It's very easy to accidentally push up or down and change the difficulty setting when picking your song. If you accidentally put it on too easy of a level, it will be literally impossible to pass the special offer. You will have to exit, and the offer will disappear. It will come back at some time randomly, but since it's random, you might not see it for a while. You might even beat the game without seeing the offer again, but this is unconfirmed.

And don't pick the wrong difficulty, or decide you want to pick a different song when you're in part 3 or 4 of the dance championship for an island. If you exit without winning, they take the whole island chunk away from you, and you have to do several stages to get back to where you were.

For freeze combo and jump combo boards, you can simply select any song while holding down "A" and set it to "All freeze" or "All Jump" accordingly. Setting to "All Jump" can also ensure high combo requests since it makes the song much easier but maintaining a high amount of steps.

You can save time by only doing the "special offers" that unlock achievements.