C.C.
For our oldest daughter's first birthday, we had a cocktail party. We figured, she won't remember it, and we were darned proud of ourselves for living through her first year! (And by then, we just really needed a stiff drink.) We invited close friends and family, and a few of them brought their kids. We had it at a local Italian restaurant - it was not exactly a "family" restaurant, but of course it was fine to bring kids there.
My recommendation is to not try and schedule any big, over the top event. There's plenty of time for that when your child turns 4 and has some clue what their birthday is all about. For the first birthday, just do what makes you happy and won't stress you out. =)