Paul's Bicycle Way of Life has rental bikes, call them to see what's available. Most of the rentals are at the 5th Street store 541-344-4105. Other than that I'm not sure who may have rental bikes.
I'd be happy to take you on a tour around town. The River trail system is near the U of O and offers 22 miles along the Willamette River. The Fern Ridge Bike Path starts at the end of 15th Street (at the fair grounds) and goes west to the West Eugene Wetlands area.