The area around Moab is incredible for star gazing into the night skies. The Milky Way is easily visible depending on clear weather, moon phase and rise, and time of year. You can check the Seeing Conditions for Moab here. Also check moonrise and phase here as moonlight can greatly diminish the stargazing.
The Bortle Scale is a measure of light pollution in the sky and very few places on earth fall into that designation. 3 parks nearby are certified as Dark Sky Parks that fall into the Bortle 2 category:
Arches National Park stargazing DIRECTIONS (Free.. Gate is unattended after 6:00pm)
Canyonlands National Park stargazing DIRECTIONS (Also free at night)
Dead Horse State Park stargazing DIRECTIONS (Probably free?)
Closer to town you can also stargaze in the Sand Flats Recreation area. Just follow an easy dirt road a few miles away from town, turn off all lights for 10ish minutes, and the sky will provide you and awesome show! SAND FLATS GATE (Free at night)