Amtrak recently changed its policies on pets and now accepts dogs weighing up to 20 pounds. There is a fee. Contact Amtrak for details. Seattle Pup encourages travelers to contact Amtrak prior to buying your ticket. Some lines may have other restrictions.
Here in Seattle the subway is called the Light Rail since it is both above and below ground and you feel like you are riding on air. Seattle has a light rail line that travels from the Sea-Tac Airport to the University of Washington. According to the Sound Transit website, small dogs in carriers may be brought on the train.
The sounder is a train that runs during the weekday to and from Lakewood and Seattle (making stops in South Tacoma, Tacoma Dome, Puyallup, Sumner, Auburn, Kent and Tukwila) and between Everett and Seattle (making stops in Mukilteo and Edmonds). Like other Sound Transit options (Light Rail and Buses) Service animals are allowed and pets may ride if they are carried in small containers.