Some Bozeman area day hikes: Bangtail Divide (you can go all the way to Swamp Creek Road if you can arrange a shuttle or are willing to stick out your thumb) – terrific views of the Crazy Mountains. Pine Creek trail to the falls, near Livingston. Hike around the Bridger Bowl Ski Area. East Boulder River trail, between Livingston and Big Timber – outstanding scenery. Bear Canyon Road trailhead, very close to town. All these are easy hikes with some wonderful vistas.
In Yellowstone Park – Slough Creek, in the northwest corner, the very essence of Yellowstone. If you're an angler, one of the best cutthroat fisheries in the world, and world-class scenery. Blacktail Creek trail from the highway down to the Yellowstone River, in its Black Canyon section. Shoshone Lake from either Lone Star trail or the Shoshone Lake trail.
The above are the tip of the iceberg – there are hundreds of great day hikes, in or out of the Park. Have a wonderful time in God's country.