Centennial Trail will give you the ability to run 10+ miles without worrying about getting lost. You can make some loops from the downtown hotels up to and around the South Hill for a solid 10+ miles, and then there is the Bloomsday course which is marked - that gives you a 12K. Riverside State Park for off-road runs.
Tracks - Gonzaga Prep, Hart Field and the track at North Central would be the best bets from downtown. Hart Field is on the South Hill and is probably the best of them. Further from downtown is Spokane Community College and Spokane Falls.
Check out the Bloomsday running club for possible races. I think you might be visiting around the Partners in Pain 5K.