Mount Douglas is a great local viewpoint and on a clear day it has a 360 degree view of the island and the Olympic Peninsula. For a good workout, you can do a short, one and a half hour hike up the mountain. The paths are poorly marked, but if you keep going up, you will find your way to the top. If you're feeling a bit lazy, you can also drive all the way to the top.
Mount Doug is close to downtown (about a 20 minute drive) and accessable by public transportation.
