Here's some basic moves -
Each character has different moves on ↓↑, ↓←, and ↑→. Those directional arrows can be switched towards whatever direction you're facing, so ↑← and ↓→ are fine too.
For each char -
↓↑ -
Marisa - drill attack up and diagonally, Marisa is invincible while she does it.
Alice - grabs an enemy that you just hit and pulls it back to Alice. If you don't press any buttons as the enemy is grabbed they're wallslammed instead.
Patchouli - Fires a whole bunch of stars. If someone is near they get whacked with all of them, or else they circle out.
↓← -
Marisa - Fires a laser. Nice on hard/lunatic because after enemies don't have invincibility.
Alice - Sends out a doll that's triggered when an enemy comes near. Theoretically could be used from crazy combos.
Patchouli - If not attacking anyone, in front of Patchouli a little ways a large attack deals damage to everyone at a slight range. If attacking someone it's focused on that person, but it doesn't lead - if they have horizontal movement, it'll likely miss.
↑→
Marisa - Lobs and exploding-magic-thing. Throws farther if you hold down attack.
Alice - Gossun, gossun... Hit z rapidly for more hits!
Patchouli - Puts the enemy in a bubble, causing them to float upwards. The bubble breaks after a short time or when the enemy is attacked, damaging everyone around it. Good for continuing chains.
Other non-attack moves-
↓C - puts slave into ambush mode. The character stops following you, and regains health at a faster rate. You can place a character somewhere to set up and advanced combo, basically.
A while being attacked - wastes a spell to push enemies that are hitting you away.
↓A next to a fainted character - wastes a spell, gives half of the active characters health to the fainted character. This is an overall win because you can put the fainted character in ambush to recover health.
That's all from memory, but I think I got it all right...