Dinosaur National Monument (DNM) is located in both Utah and Colorado - the Utah side is just outside of Vernal, Utah and offers a large dinosaur quarry, trails, camping, and amazing scenery of the Uintah Basin and Green River. Petroglyphs hint at earlier cultures (mainly the Fremont), and later homesteaders and outlaws found refuge here. Of course the name of the game here are the dinosaur fossils. Visitors can see over 1,500 dinosaur fossils exposed on the cliff face, located inside the Quarry Exhibit Hall. It's one of the most amazing sites for Paleontologists and explorers-alike. Plan to spend at least one full day at DNM.
Spooky & Peekaboo are the most popular slot canyons in the Escalante area. The access is easy, you can easily do both as a loop in a few hours, and hikers without technical canyoneering equipment or skills can enjoy both. Spooky slot canyon is one of the narrowest canyons in the world that hikers can safely navigate. It's much longer and slimmer than Peekaboo, forcing hikers to carry daypacks to the side as you shuffle through. Add in a few short obstacles and one 5 ft scramble to make it that much more fun. Claustrophobic people will have a hard time with this Spooky.
The Rochester Panel and Molen Reef Petroglyphs are near Emery, Utah, and consists of one very large panel of hundreds of images likely carved by the Fremont Native Americans. The vast majority of the petroglyphs have a strong Barrier Canyon influence, including animals, female/male figures, and one large rainbow-like image. Molen Reef Petroglyphs are much smaller, but still expose what the native people were contemplating. You can easily combine both in one afternoon, as the hike to the Rochester Panel is only 1/2 mile, and you can drive right up to the Molen Reef area. Kids, dogs, and casual hikers will enjoy these areas, but please respect the rules and DO NOT touch any of the petroglyphs.
The Dutchman Arch & the Head of Sinbad Pictographs are located in the San Rafael Swell - they are actually only about 1/4 and 1/2 mile north of I-70, but driving there takes some effort and a 4X4 drive car. Dutchman Arch is actually quite large, and a fun area to explore with dogs or kids. There's no hiking involved - you can park on either side of the arch.