The Bayan-Ölgii Museum is the best first stop in town. Its displays are old-fashioned, but they give useful context for a province where most people are ethnic Kazakh and where eagle hunting, embroidery, Islam, and nomadic herding all sit beside Mongolian state culture.
Look for the costumes, hunting gear, musical instruments, and local wildlife displays. It is a small museum rather than a major institution, but in Ölgii it answers questions you will have as soon as you step outside.