I've spent a fair amount of time exploring Bears Ears NM. There are basically three areas that most people visit. Lower Butler Wash (Comb Ridge), Cedar Mesa & the area up on Elk Ridge (through the Bears Ears formation). As Bill noted the upper elevations are probably still muddy from the snow this year. Assuming you are doing day hikes & have a high clearance vehicle this would be my list. All of these are pretty well documented on the internet, but I'm happy to share some sources, actual tracks, & waypoints if needed,
Lower Butler Wash Road (262) runs between Hwy 95 & 163. There are numerous canyons that extend up into Comb Ridge that can be easily done as day hikes. The most popular are Monarch Cave & Procession Panel, but there are many others.
Day hikes on Cedar Mesa: Kane Gulch (ranger station to Grand Gulch), Moonhouse (needs an online permit reservation), Fallen Roof & The Citadel (same side road), Bullet Canyon, Mule Canyon, Cave Towers, Butler Wash (off Hwy95). And Natural Bridges NM - not part of Bears Ears, but worth the visit since it's on Cedar Mesa.
You can camp anywhere on BLM land. Butler Wash Road has many turn-offs to the various trailheads & camp spots. On Cedar Mesa it's the same once you leave Highway 261 or 95.
If you go to Bluff stop at the Bears Ears Education Center. On Cedar Mesa, the Kane Gulch Ranger Station is open 8-12 every day until June 15th & they are very helpful about road conditions, etc. Natural Bridges has water, trash disposal & a campground if you need it. I usually shop in Blanding for groceries & gas (closed on Sundays).
This is the BLM website for day hiking passes & other information.
Have a great trip. Rick