Lots of great near-by climbing in Seattle for all levels of climbers. The diffenitive guide to my favorite crag is the Exit 38 (Deception Crags) Rock Climbing Guide
There is also Exit 32 for the more advanced climbers.
There are also a few indoor rock climbing gyms where you can go to learn and train: Vertical World and Stone Gardens. The University of Washington also has an indoor climbing wall for students, faculty and staff.