Seattle now offers light rail connectivity from the airport to several stations downtown and now even Capitol Hill & University of Washington. One-way fare to or from the airport & downtown is $3.00. There are ticket machines at the stations. Also see below for information about the ORCA card. The travel time between airport and downtown is about 45 minutes.
Seattle now allows for Lyft and Uber pickup at the airport. There is a designated location located in the bottom of the parking garage for ride-share pickups. Ride-share pickups are not allowed outside the baggage claim. Rides TO the airport from anywhere in the city are allowed though and uber / lyft drivers are all quite familiar with this.
From rental cars to limos to busses to shuttles, SeaTac offers several other ground transportation options as well. (Note rental cars are now ALL off site at a seperate facility available via shuttle.)
http://www.portseattle.org/Sea-Tac/Parking-and-Transportation/Ground-Transportation/Pages/default.aspx
Seattle has vast coverage of the city by Bus, Light Rail, and street car. The Seattle Transit Map offers a great interactive way to browse different routes.
Lyft and Uber are very popular services in Seattle. If you have an account, it will work well here. We use this often for travel around the city.
Other options we have are car sharing services: Car2Go and ReachNow (BMW). With these you rent cars by the minute and leave them in any official Seattle parking spot including meters for free. You are only charged for the distance traveled and time taken and can simply sign out when you get to your destination.
Here in Seattle, all transit options (rail, streetcar, bus, even Ferry) use the ORCA card. These are available for 5$ plus whatever funds you want to add to it at the airport as well as Westlake station downtown. Just use the ticket machines to purchase. These cards are also handy if you ever want to transfer from one service to another (buses offer 90mins to transfer to another line for free). They are not required, but offer an easy way to pay for transit compared to cash / exact change.
Also available are daily passes on any transit option as well.