a lot of times you can play it like you would any other hand (a pocket pair or whatever), some times you might slowplay it a bit. depends on a lot of things like who you're playing and position and what the blinds are, etc.

on the flop, you have the nuts. the nuts is the best possible hand at the time. on the river, aaaa could possibly be the nuts, but it doesn't exist yet, you still have the nuts for now.

the turn and river actually could bring many cards which would make you lose, there are lots of different runner runner straight flushes that could beat you, that is if you lived in hell.