it depends if you want to set up a tent or are just car bivying. If you're doing the car bivy, check out the gravel parking lot on the corner of Merman Dr and Valley Rd, near the athletic playfields. Alternately, you can drive down Park St for about a mile to a pullout by the Humane Society shelter. less traffic there for sure. If tenting it, that's a bit harder...I'd probably drive east of town on Hwy 27 towards Palouse and hit one of the dirt roads. expect to get up early if a farmer finds you sleeping in his wheat fields. As for bathing and stuff, you can get a day pass to the Student Rec Center for $8 (or go with a student who "sponsors" you and pay $5).
Get a beer at the Sports Page, downtown. Rico's (across the street) has the best fries in town.
if you want get get some climbing in, check out this link: http://myrso.wsu.edu/wsualpine/index.html