Get the city map from the tourist office and if the Deer Mountain Trail (3 - 5 hours) seems too long/difficult, try this one.  It's a very steep walk up the street to the get to the trail head and the first part of this 45 min. hike is a bit steep on the edge of the cliff, but it's not that bad. You can go to the end (University of Southern Alaska) or cut it short and take the first set of stairs down, and the walk back into town, if that's where you started.  Some really nice views and it's really nice to get out of town.