Depends on the character (I have many favorites):
My Fursona: I'd probably greet him cordially like I do with anyone I meet. I'd then start to talk to him about other universes and the like, then I'd probably join him in his travels after getting a little bit more combat training.
Other OCs (Joshua Tiger): *blushes* I really don't think I could say that here...
Character's from Entertainment (Derpy Hooves): Well, first I'd take a couple of pictures, for proof that this actually happened. Then I would interview her, asking her a few questions regarding her home. Then... I really don't know... Go get some coffee at Starbucks or something? I'd probably hang out with her often, that's for sure.