There are 13 requests in total and the final one is given to you by Wise Man. To complete a request you just need to get your katamari to the size that King tells you within the time limit, your score does not matter. None of the requests are very troublesome and you will probably complete them all within two or three retries. If you are having difficulty though, the following tips may help:
Remember what order you can roll things up in if you are retrying the request. There is no point wasting time trying to roll up an item that is too big.
Don't waste time with curio's - leave them until eternal mode as that will have no time limit or goals.
In requests with more than one area, when your katamari is big enough to move on, don't! Stay in the same area a little longer to make the katamari a small bit bigger. This is because you can generally roll up almost everything in the current area when you're big enough to move on, but everything in the next are will be do big you will only be able to roll up basic items. I usually wait roughly 30 seconds before moving on as that's usually enough time to give the area a quick, final sweep.
Stretch your katamari! If you're trying to pick up a bunch of small items, e.g. thumbtacks, then stretch your katamari so you can pick everything up with one roll instead of wasting time rolling around on them. You can stretch your katamari by putting two fingers on the screen or rear touch pad and separating them. You can shrink it back to it regular size by pinching your fingers back together.