I go down there a few times a year to mountain bike. It rarely rains, but bring a light jacket. Definitely bring shorts and shit for the day because it gets hot as fuck. Also bring a few layers for night because the desert cools off when the sun goes down.
If you're bringing your own bike, bring a shit ton of extra tubes, and stay on the trails. There are these little bur-like things everywhere off-trail called "goat heads" that will pop any tire. Bring some bomb shoes for hiking and a camelback or some shit because you'll get mad thirsty walking around in the sand all day.
Other than that, jsut bring your normal camping shit. If you're trying to find free camping, you may be a little way from town, but there is cheap camping in the Sand Flats BLM park just outside of town.
Here is Moab's site's page on camping around town.
http://www.discovermoab.com/campgrounds.htm
Good luck... and even if you have shitty luck it'll still be fucking fun.
Oh. And ride the trail, Porcupine Rim. It's not too hard but it has got a full range of terrain and it only takes about half a day. It's about 18 or 19 miles. You'll need to pay for a shuttle up to the top, but it's worth it.