There are many adventures that you can choose from, and each character has their own unique one. The three I used were Mr. Incredible's, Sumo, and Castles & Slingshots.
In Mr. Incredible, you need to destroy 3 domes before the clock runs out. Each dome has a number on it that indicates how many times you need to hit it. When enemies swarm you, use a ground pound (A, Y) and knock them back, then hit the dome. For bigger Omnidroids, move around to the other side of the dome if you can.
In Sumo, you need to stay on a platform of blocks for 3 minutes. Enemies of all kinds will swarm you, but just use one move, over and over again: ground pound. It keeps you in the air out of the line of fire, and knocks enemies off the edges. Just be careful toward the end, because there won't be many blocks left, and you can land right on one that's disappearing.
Edit: Chimpotle noted that if you play as Violet (not included) you can go invisible and camp in the middle. Nice!
In Castles & Slingshots, there are many balls hidden behind 3 castles made out of Legoesque bricks. You need to hit 3 with your slingshot in a minute. This is kinda luck based, but if you do it enough, you'll stumble on them. When I got gold in this, the 3 balls were literally lined up next to each other on the same platform (I finished the mission with 5 hits). The key is rapid fire. Don't wait for the shot to hit, just keep pulling your LS back and flinging them. You'll need to throw a bit farther at times, of course, but you can clean up a lot of blocks at the beginning by just flinging lots of shots quickly.