Matt, I've biked just a few places around Jackson, but here are some possibilities. Right out of Jackson is Cache Creek, a well-used (hike, bike, horse) trail that dead ends at the Gros Ventre (I think) wilderness area. It's got a nice, gradual grade, with a few techy spots where you cross some drainage. For a bit larger workout, after about two miles or so of riding, look for the trail heading to the right (I think it's marked). This trail takes a short, steep climb up the ridge, then tops out and descends to Game Creek. The first part of the descent is steep and dry with loose gravel and switchbacks, but once you get off the ridge, the riding is quite easy and fast. The trail comes out onto the road which takes you out Game Creek where you join Highway 89 south of Jackson. Across the highway is the start/end of a biking path that will take you all the way back into Jackson. I nice loop.
Of course, Snow King has many trails, all pretty steep, and lift service. Another easy road is the one leaving northeast Jackson and skirting the east side of the Elk Refuge. The road then ascends the hills to some National Forest land. There are also a number of National Forest trails/roads leaving from the Gros Ventre Canyon area near Slide Lake. Haven't any experience with those. Check local bike shops.
Snake River Canyon has some great steep single track heading up various canyons: Cabin Creek, East Table, Cottonwood, etc.
Teton National Park does have a few roads to bike on. One that's off the beaten path is the old RKO Road, although I believe it's referred to by a different name in the park. It starts/ends(unmarked) just past the Cottonwood Creek picnic site and heads out toward the SNake River, then follows along the west bank of the river, coming back out to the inside highway near Signal Mountain. Very cobbley and rough, but a fantastic ride for animal viewing. You could do this as a loop by riding back on the highway. A nice, paved loop could include riding from Jenny Lake up the highway to the turnoff to String Lake, then back on the one-way road along the shore of Jenny Lake.
Off course, these are just the few that I'm a bit familiar with. Best bet would be to contact one of the bike shops in town for a map and some more accurate info.