Yeah, I thought of another one...
Great Basin National Park is just over the border in Nevada, just outside Baker. Not a fantastic location but much closer at about 4 hours. I enjoyed the "Alpine Lakes Loop" hike and Lehman Caves. Lexington Arch is in the park but a short drive away down a dirt road. The only place to stay in Baker is the Silver Jack Motel, if I remember correctly, and it has only 4 or 5 rooms.
Yellowstone and the Tetons are only about 5 hours away the other direction, too.
Vail Valley, Colorado the surrounding area (Leadville, Aspen, Snowmass, Estes Park) are about 7 hours east.
Wow, I hadn't really thought about great a location Salt Lake City is before, but it's fairly central to a lot of absolutely beautiful landscapes. I hope you have a lot of weekends free...