There's lot's of cycling around here, whether you're into road or mountain or both. If you're interested in events, you could check out this calander of races and other rides. http://www.spokanerocketvelo.com/races.html
The page gives a comprehensive look at the formal, organized rides in the area. If you're interested in less formal rides or routes, there are plenty of places to ride. Bring both your mtn bike and road bike.