Anchorage works best as a base — the city has a few specific draws but most of what makes Alaska worth visiting lies just outside. Spend half a day on the downtown waterfront and museums, then use the rest of your time on day trips or to gear up for a longer trip into the interior.
The Anchorage Museum is the best single attraction in the city and the right introduction to the state. The Tony Knowles Coastal Trail is the best free thing to do — 11 paved miles along Cook Inlet with regular moose sightings and clear-day views of Denali. The Alaska Native Heritage Center covers the Native cultures of the state at a depth no other museum attempts. For families, the Alaska Zoo houses bears, lynx, and the only polar bears in the state on public display.