Make sure you take the Hartnett road ( if you have high clearance, you have to cross the Fremont River on that route and need 4x4. Water very high this weekend..) The Cathedral Valley road was easy and most cars can get you there without AWD or 4x4.
I made some mistakes of not checking all my plan details before resulted in wasting a day or so, so hopefully I can help you avoid that!
I trusted Google maps too much and it took me to the route on the far west side ( by Fremont and loa) which altered my route and ended up making me drive 2-3 hours more than needed. The access route it takes you to from Salt Lake is the hardest route to get up and is closed currently. It's quickest from salt lake to go through price then get off the hanksville exit , drive through hanksville and in about 15-20 minutes while driving east towards Torrey get off the cainville wash road and connect to the cathedral road.
Camping is available all over the outside boundaries of the park which is definitely better to stay before the sun tower and moon tower (if those are your main goals of places to see which is for most.)
I made the mistake of getting their after dark and going all the way past the towers to the camping area to the only designated "campground" that only held 8 people and was 60-90 minutes to drive through rough washboard dirt roads and climbing a rocky hill.
There was Nothing more special about the campground that is any different than finding your own site before the park. It's a good place if you intend to visit the overlook area on the far west side of cathedral Valley but if not find your own spot.
No water sources, or trash bins.
PACK IT IN, PACK IT OUT, PUT OUT YOUR FIRE 110% AND LEAVE NO TRACE!