Is there some kind of way that I can make meshes not pass through each other when I'm animating them? Like maybe use rigid bodies or physics or something like that (sorry, i'm still new to Blender)? Like for example, if I want two people to handshake, how do I make it so that the hands don't go through each other? Or do I just have to animate it in such a way so that they don't?
You can use the Soft body modifier after you animated the general movement.