The North Entrance through Gardiner is the closest to Bozeman (approx. 80 miles, 1.5-2 hours). This provides direct access to Mammoth Hot Springs. The West Entrance in West Yellowstone is slightly further (approx. 90-97 miles, 2 hours). This provides access for reaching geyser basin (Old Faithful).
The Museum of the Rockies is a research and history museum, famous for its collection of North American dinosaur fossils. It features the Siebel Dinosaur Complex and a planetarium. Great for the kiddos (and adults) ;)