I just moved into an older apartment, and the bathtub constantly drips and when the water is on the water comes out from the handles. I'm afraid to mess with it because while I was cleaning the tub after I moved in, my thumb punched a hole in it around the handles and the caulk won't hold. I barely touched it! What should I do about either problem?

It sounds like you have three problems.
1. The washers and seats in the tub faucet need to be replace to stop the driping faucet.

2. The bonnet or packing nut around the stems needs to be tighten to keep the water
From coming out around the handle.

3 Repairs need to be made to the wall to correct the damage done by the water leaking
Out around the handles and the tiles reset and sealed.

All of these are the responsibility of the landlord.

"What should I do about either problem" well, you should call your landlord. Sure we can help people fix problems like this and Harold gave you some good advice. However, in a rental you need to consider what happens if you try unauthoized repairs and you break things worse. I'd call the LL or his maintenance people first before trying anything on your own.