I literally just asked a ranger this question last week. She said show up at 7 or 8am and get on the list for sites to become available. The sites have to be vacated by noon but I forgot to ask if people can reserve them for multiple days and just get them from say a Friday through Sunday. She said that if you get there early and are one of the first to sign up for the next available site, you usually get one. Check all the campgrounds in the park to see if any one has plain old reservations available. Also, don't forget that you can backcountry camp for free with a permit. This means that you just have to find a site that is x number of feet from the trail and bodies of water. There are also a few other regulations so check all of the rules when you go get the permit. I believe some areas may require a bear canister. Good luck and enjoy the park!
It is pretty hard to determine and depends on the campground, weather, and date. I use Lewis Mtn campground the most (it's small with only 31 campsites) and the best ones are gone by Thursday afternoons so i get there by 1 pm. I enjoy being there early and getting an extra night in for the weekend.
Can't answer any better than brian and david did.
David Zehe's answer is right on !
When you enter the park they have a sign saying if the Campgrounds are full or have availability. I imagine that the visitor's centers have access to this info as well. However just because it says the have availability or don't, doesn't mean that they won't later in the day. There are plenty of campgrounds with plenty of sites, i wouldn't think you would need to arrive early. You can also reserve online if you want to be absolutely sure.
Thanks! Your answer is awaiting moderation.