Auburn is a car-dependent city by design. The historic downtown and the Auburn Station area are walkable enough, but most attractions are spread across the city and require driving. King County Metro runs bus routes through Auburn connecting to the Sounder station and nearby communities.
The Green River Trail is the best way to get around under human power — it connects several of the more interesting areas of the city along the river corridor and is well maintained for cycling. Bike rentals are not readily available in Auburn itself, so bring your own if cycling is the plan.