We left Goblin Valley State Park and drove to the city of Green River (population about 900). Seriously small stuff. What's in Green River? I've heard they have the best watermelons around. So I bought some.
What next? Green River State Park is mostly a camping site for those rafting the River. No hikes or anything exciting. We were tired so we decided to stay the night, but not in the state park.
The best free camping spot...Swasey's Beach. (Not THAT Swayze!)
This spot is designated as recreational access with no permits or fees for camping or boating. The sand is nice and soft to sleep on and the water might be the brownest/muddiest water you've ever seen (but still great to jump in to cool off). Be aware of the strong current. This is also a great place for a wonderful view of the stars in the sky at night.
Go East on Mail Street. Turn left (North) at Hastings Road. Look to the left at mile 2.4, you'll see a marker where the Spanish Trail once crossed the River where it was the only place to safely cross for hundreds of miles. The beach is at mile 9.5. The road is paved all of the way except for the last hundred yards or so.
If you keep going on the dirt road (it gets pretty rocky/bumpy), you can see the Old Stone House at mile 12 (many say it's haunted), petroglyphs located on a low rock ledge at mile 17.8 (depictions of big horn sheep, elk, and snakes), and at 18 miles, look towards the south to see the famous Nefertiti formation. That's the end of the road. From there, you may hike, ride horseback, or bike further into the Lower Gray Canyon, but you have to return the way you came to go back to town.