If you don't know where to begin, try a nice local bike shop like Wheelworks on W11th at Lawrence or, if you want to go upscale, Arriving by Bike will on Willamette will help you out. Lifecycles at Pearl and 17th is very good as well. If you think an e-bike may be just the ticket for you, there's an entire shop dedicated to just e-bikes (Eugene Electric Bicycles) on High St near 5th (behind the building).

All of these shops are run by very nice people who know what they're doing and all are locally owned and operated. 

If you are going to do the walking thing, consider Baker Boots at four corners. Our sidewalks and bike paths aren't exactly sneaker-ready (unless you like walking in eight-inch puddles).