Bike the Route of the Hiawatha railway, a gentle 15 mile downhill ride described as one of the most scenic stretches of railway in the country. Your path includes 9 train tunnels (including the 1.6 mile Taft Tunnel), 9 high trestles. and, best of all, a shuttle bus at the bottom to take you back to the top. Trail and shuttle tickets, bike , bike rack, and helmet rentals are available at the Lookout Pass Ski lodge on Interstate 90, only 100 miles from Missoula. 12 miles beyond is the historic mining town of Wallace. There are a number of trailheads, but the East Portal is the most popular, and highest.
The TransAmerica Bike Trail, a 4,250 mile route, runs 75 miles through the Bitterroot, from Lost Trail Pass to Florence, following the Lewis and Clark Trail. For a map and practical advice, check out these two sites—Adventurecycling and TransAmericaTrailCycle (this site is from 2008 but has good advice for the biker).
Another option is to take off on one of the side roads, find a place to park, unload your bike, and enjoy riding along the back country routes through green fields with grazing cattle and sheep.