All you have to do is simply get the item Elizabeth wants or kill a certain enemy that Elizabeth wants you to kill and bring it back to her to receive a prize.
Prizes can range from Yen to items to very good pieces of equipment early on in the game.
When you fuse, the end result will often inherit skills from either one or both of the Personas you're using.
For example, let's say you're doing this: Persona 1 Persona 2 --------- --------- Agi Zio Bufu Bufu Tarunda Tarakaja There's a chance that the end result will inherit one of those skills if it doesn't already learn it naturally or doesn't already have it in the future.
There are no random battles in this game, so you have to touch a shadow to initiate a battle.