What kind of climbing? Rock or mountain?
There are several classes available for both. You can learn to rock climb at the local climbing gyms (Stone Gardens or Vertical World). REI also has classes.
For mountaineering, you can try the Mountaineers, Alpine Ascent International, Raineer Mountaineering Inc, Washington Alpine Club, Mountain Madness, or many others.
Let me warn you, if you want to get into mountaineering right now, most of the classes are closed for registration. RMI, Mountain Madness, and AAI have classes open in the summer for a large fee.