I don't know about Moab since I took my own Jeep there, but the same rental companies operate out of Ouray, CO. There they have strict limits on which trails you can take the rentals, specifically excluding almost any but the beginner/novice trails. I wouldn't be surprised if they had the same limitations in Moah.
The trails they will let you run are a bit more than the novice trails. They have some very nicely equipped/lifted jeeps in Moab to rent. One thing to remember though is you pay for ALL damage and BREAKAGE.
However, before you rent, you may want to just explore in your truck, even if it is 2wd. You can enter at the Pot Ash roads and do some of the White Rim Trail, explore the gravel roads that go around the north end of Arches, etc. I did all of these with a long bed crew cab 4x4 92' Ford truck and didn't use 4wd on any of them.
If 4wd, you can also do the trail almost to Gemini Bridges, Shaffer Trail, Long Canyon, and a bunch of other trails. These I also did, but did have to use 4wd a time or two, but nothing serious.
I'll also suggest a jet boat ride on the river as it is cooler down there.
Just a word of caution, make sure someone know where you are going and when to expect you back. Lots of the trails don't have cell coverage due to the topography.
2015 Ram 3500 4x4 Crew Cab SRW 6.4 Hemi LB 3.73 (12.4 hand calc avg mpg after 92,000 miles with camper)
2004 Lance 815 (prev: 2004 FW 35'; 1994 TT 30'; Tents)