You should check out the Native Heritage Center, it's toward the edge of town on N. Muldoon Rd. Then there is the downtown museum near A St. between 6th and 7th. There's a trolley ride that goes around town and gives an informative tour, downtown market on Saturdays on 3rd., the Alaska zoo on O'Malley (you can see the wolf pups that were rescued earlier this summer by the fireman fighting a major fire on the Kenai peninsula), or a trip out of town to Portage Glacier, you can take a boat ride out to the face of the glacier and watch it calve. Downtown Anchorage has lots of places to eat, but 3 of my favorite are The Bridge, down near the railroad. It spans a creek and you can enjoy good food and watch the fisherman fishing in downtown. Humpy's on 6th is good and The Snow Goose on 3rd are also great places to eat.
Mostly my best advice is to dress in layers, polar fleece and raincoat, and don't let the weather deter you from enjoying what's out there.