Auburn is on the Sounder South commuter rail line, with multiple trains daily to and from Seattle King Street Station (about 40 minutes) and Tacoma (about 25 minutes). The Auburn Station is in the heart of downtown and makes the city genuinely accessible by rail from both cities — a useful fact if you're combining Auburn with a day trip.
By car, Auburn sits at the junction of Highway 167 (the Valley Freeway, running north-south through the Green River Valley) and Auburn Way, the old highway. Seattle is about 30 miles north, Tacoma about 20 miles south. The drive times are heavily dependent on traffic — allow at least an hour to Seattle in rush hour. Sea-Tac Airport is about 12 miles north via Highway 167.