The obvious day trip is Yellowstone itself — the northeast entrance at Cooke City is 65 miles up the Beartooth Highway, around two and a half hours' drive. The Lamar Valley, ten minutes inside the park gate, is famous for wolves at dawn and dusk. Pack lunch, leave early, fill the fuel tank in town before you set off.
Billings, an hour to the northeast, is the nearest real city for shopping, airport runs, or a hospital. It is not a destination in itself, but the Yellowstone County Museum and the Pictograph Cave State Park on the southeast edge of town reward an afternoon.
Bighorn Canyon National Recreation Area, two hours southeast, offers a striking high-walled canyon, boat trips on Bighorn Lake, and a quieter alternative to Yellowstone for hiking. The detour through the Pryor Mountains, on the way out, passes wild horse range — one of the few free-roaming herds left in the United States.