Rainbow Bridge Trail is a 32.7 mile moderately trafficked out and back trail located near Monument Valley, Utah that features beautiful wild flowers and is rated as moderate. The trail is primarily used for hiking, walking, nature trips, and backpacking and is accessible from March until November.
My then-girlfriend (now wife) and I hiked the South Trail into Rainbow Bridge. (The map on this site shows the North Trail.) It was a truly life-changing experience. We didn't see another soul over our 3 days on the trail. Be careful: the first water is at the 9-mile mark. Also, there's a lot of elevation change. You work your way in and out of steep canyons for the first 6 miles. The NPS link I pasted below has a pretty good description of the trail. I'd recommend that you ask one of the local Navajo if you can park on their property. They'll keep your vehicle safe. We parked at Marlo's house and paid $20 as a courtesy. He and his wife were very friendly.