Mountain Biking
The Denver to Jackson route is a pretty good one. You can cut the corner of I25 and I80 by taking a US hwy at Ft. Collins(last Waffle House for a while).
Lots of good gravel and sand roads outside of Rock Springs, WY but it's an oil town and flatlanders stick out pretty easy there.
If you're looking for mountain biking. The Teton Valley in Idaho has a great community of bikers. It's about 30 mi west of Jackson. Habitat High Altitude Provisions in Driggs, Idaho can help you out with great local knowledge, equipment and maybe even a beer at the end of the day. One of the owners, Mitch, drives an FJ-40 when he's not hauling a drift boat or a truck full of bikes.
Speaking of drift boats, if you're into fishing, do it. It's awesome out there.
May is a tough time, it's mud season and a lot of snow is melting full bore but it's still worth going. You may be able to get into and out of Jackson and Yellowstone OK but a LOT of people and RVs move into Jackson and Yellowstone. It can get nutty.
Good luck. Hope you have lots of time out there. Best place in the world.