Try the Steel Bender trail, it begins at Ken's lake south of town and follows the creek for a ways. Once back in there, you should be able to relax and cast a line. The Colorado and the Green are pretty big rivers and are likely flowing high right now, I don't know what it's like to fish those. (I shoot stuff, I don't fish.)
No trout really around Moab unless you find some in mill creek in the La Sals. Mill creek is pretty small, not sure if there are trout, but there should be higher up at elevation. The Colorado and the Green are in full flow, and too murky and warm for trout. My fishing equipment saw no action in Eastern Utah last June. Awesome fishing in north Utah near flaming gorge dam, but a long drive away. Highly suggest leaving Moab and heading north if you truly want to chase trout.
Lots of pay camping along the River road highway, some free stuff here and there away from the river. Camping in the La Sals offers cooler temps, shade, and some free camping. I'd suggest the La Sal loop road this time of year or go up to La Sal pass.