Yes, all of this. I'm not hugely prescriptive, so I'd definitely give some weight to "whatever works for your family." But having run across grown women who can't bring themselves to say more than "down there," I think there's a lot to be said for learning to use the right names from the beginning.

ETA: the specific terms our family uses are penis, vulva, and (inconsistently, I guess) bottom/butt.