Portland International Airport: A rental car is the best choice if the Fruit Loop, trailheads, Mt. Hood, or Gorge waterfalls are part of the trip: Hood River is roughly 1¼–1½ hours away via I-84, traffic permitting. Without a car, take TriMet MAX Red Line to Gateway Transit Center, then the Columbia Gorge Express to Hood River; allow about 2 hours total and roughly $13 ($10 for the Gorge Express plus a TriMet fare). The Gorge Express runs only about four times daily, so this is a schedule-led transfer, not an airport shuttle you can improvise.
Bingen–White Salmon Amtrak station, Washington: The Empire Builder from Portland reaches Bingen in about 90 minutes; coach fares can be as low as about $7. On weekdays, connect with Mt. Adams Transportation Services to Hood River, generally about hourly, for $1. This is a very good low-cost option only if its limited timetable lines up; it is not dependable for a weekend arrival.