There will be some sort of epic adventure within 5 miles. You can not go wrong. Some great discussion here. A few things about walking around out here:
1) Footwear. I have hiked for years down there with Chacos. Avoid big heavy boots. Trail runners are great.
2) Clothing. Full coverage and light is the key, unless you like being smothered for days in Deet and sun goop. Wide brim hat is almost required. Permithrin has changed everything for me.
3) Check the forecasts. Stay out of narrow canyons (or be very careful) if it's supposed to rain in your chosen river basin, not just where you are. I have been snowed on in June in southern Utah, but May is just great down there generally. Monsoon season usually kicks in at the end of June, but T-storms can happen any time. You have not seen anything in this world until you see (feel) a major flash flood roll by. Very humbling.
4) Vehicle. Know that many remote areas, including in the parks, require high clearance vehicles and a means of extraction. 4x4 makes me feel better. If you are renting, I would stay out of the Maze, Kaiparowitz Plateau, etc.
5) Water. Use local knowledge of sources. Year to year, water sources vary wildly. Talk to rangers, hikers as you pass, etc. If in doubt, have extra carrying capacity. I have been known to carry a collapsible 5 gallon container if I'm really out there.
Enjoy! You are welcome to PM me (I'm in SLC) and talk about specific areas.