Train
When you first get to Seattle, go ahead and get yourself an Orca card. It's a plastic card that you can keep refilling via the Orca app on your phone. It works for both the Sound Transit trains and buses. Travelers with carry-ons are always coming to and from SeaTac airport. So don't feel weird about taking the train with luggage. We've never really had a problem on public transportation here. It's been clean every time we've been on it. There's never a lot of waiting for the train either. It's pretty great!
Flight
SeaTac (named because it's in between Seattle and Tacoma, WA) will be your best bet. DYK: If you search Google Flights, you can track prices and get alerted when fares are below average. Check it out google.com/travel/flights I've also used the Hopper app for the same purpose (not to actually book flights, but helpful to know if it's a good time to book).
Rental Car
I noticed that some rental car places (like Enterprise, Hertz) are closed for Memorial Day, Monday 5/26, EXCEPT the SeaTac airport locations. So, a rental car may or may not work out depending on when you're leaving. But, if you are going to try to get a rental car, it's probably best to book far in advance. Like I said before though, the 1 Line is great. Pretty much will get you everywhere you need to go, and if not, the bus can connect you. :)
Travel Note
DYK: On Google Maps you can change Driving directions to Walking directions, Public Transportation directions, and more. Very helpful for navigating! Oh, and in Seattle, whatever the Google Maps ETA says, best to add 30 minutes to be on the safe side. :)