At Anchorage Museum of History and Art you find everything about culture and history in Alaska and the surrounding areas.
One place in Anchorage I recommended was Cook Inlet Book Company - a bookstore specialized in books about Alaska. You find books about Alaska you cannot get anywhere. You can also hear writers from Alaska reading samples from their books. But it's closed now!
Anchorage is a city where you should take long walks. In sommer after dinner there is enough daylight to walk around for hours. To reach places even faster I recommend People Mover. In the downtown area it's free to take the bus ("Downtown Anchorage Short Hop").
Be sure to visit the Imaginarium - I hope to see it myself someday (I like their website).