To explore most of Washington outside of main cities, you will generally need a car because there is very little public transportation, especially to more remote locations.
By train
Getting around by train on Amtrak (http://www.amtrak.com) is likely to be quite a hassle, especially with infrequent departures, slow travel times, inconvienent schedules and limited routes. For those who still wish to take trains to get around Washington, these are the main routes.
Amtrak Cascades: Goes along the I-5 corridor, connecting Bellingham, Everett, Seattle, Tacoma, Olympia and Vancouver, WA together with Portland, OR and Vancouver, BC.
Empire Builder Seattle branch: Connects Seattle and Spokane through Wenatchee.
Empire Builder Portland Branch: Connects Portland and Spokane through the Columbia gorge and Pasco.
By bus
Greyhound Bus Company (http://www.greyhound.com) has the largest bus route system in the state, mostly along the Interstates 5 and 90. In addition, most counties have their own bus service, but this makes going long distances through many counties a hassle. See the individual region and city article for more information on public transportation.