The Rattlesnake National Recreation Area, lying directly north of town, is what most people think of when they think of mountain biking in Missoula. A wilderness feel, smooth singletrack, and, it's close enough to ride from downtown.
The Rattlesnake NRA was created by the same act of Congress that created the Rattlesnake Wilderness just to the north. While mostly known for smooth trails, Overlook and the Stuart Peak Trail/Upper Spring Gulch have some fun, steeper technical sections. Sawmill Gulch is where you'll find some mellower riding and gentler grades, and on your way back from there make sure not to miss a lap on Ewok, which parallels Sawmill Gulch Road.
Dogs are not allowed in Sawmill Gulch (Fenceline) year-round, but during the summer season dogs are allowed in most other areas. Check the signs at the trailhead.