If you like the desert, visit the Moab area and South-Eastern Utah. Canyonlands National Park, Arches National Monument, and some places outside the national parks that are very remote, at least for the US.
If you prefer alpine environments, visit South Western Colorado. The San Juan Mountains are some of the most beautiful anywhere. Visit Ouray, Silverton, Telluride, Lake City and others.
If you enjoy going off road, there are plenty of opportunities in both locations.