Link Light Rail Website:

In general, to go north and south for a distance, the light rail is the best option. It is also your cheapest option for getting to and from the airport, $3 each way.

https://www.soundtransit.org/ride-with-us/stations/link-light-rail-stations

Bus Service Website:

For shorter trips and to go east and west, you need to take a bus.

https://share.google/YooFUBdEnv6FautKJ

Uber/Lyft/Taxis

These are quick and plentiful but expensive. Seattle is the most expensive city in the US to take an Uber. An Uber from the airport to downtown will be about $65-90. An Uber from the Mayflower or the Hotel Andra to the wedding venue will be about $12 each way so plan ahead and share rides.