Games : Am I really supposed to just wander around?

So I found the Moon's Tear and did some other stuff and now the game is just like "walk around! talk to people! kill time!"

Am I really just supposed to waste whatever time I have until the Clock Tower opens? It's boring, and I feel like I should be accomplishing something.

Should I just wander aimlessly? Or are there ways to spend my time more productively?