I lived in Spokane for a few years not too long ago, great place for railfanning. Lots and lots of abandoned lines, historic artifacts and locations, etc. A lot of really great info's already been posted by others...
If you're into chasing trains, I was always partial to the UP myself. AFAIK, it's still track-warrant territory from well south of Spokane all the way north. Easy to find and follow trains with a scanner, though track speeds have been increasing steadily since I moved away. When I was there (2002-2004) the UP had a great branchline feel. Huge trains w/ interesting power on not always very good track, great scenery (especially between Spokane and Eastport),some slow running, "interesting" meets, etc. UP crews were always very friendly, and a surprising amount of the UP line is easily accessible via local roads, especially south of Sandpoint, ID. A lot of southbounds changed crews at Felts Field, don't know if they still do but it was a good place to chat with the crews, get roster shots and there's a great cafe in the airport there.
Maybe it's already been mentioned, but a little swing through Marshall Canyon is a must. I'll let someone else explain the history, it's complex. Lot of action, though. Just be careful, it can be a fast highway and some of the side roads turn into BNSF access roads pretty quick. Not such a hot idea (at least I don't think it is) to venture too far up those.
I enjoyed Spokane a lot, haven't made it back in a few years but I'm hoping to soon.
Have fun.