Redmond is a crappy place to be without a car of any sort. The only reasonable mass transit around there is an infrequent bus system designed mostly to get employees to and from Microsoft. It can take 2 hours via public transit. Taxis are not cruising the streets so you have to call them well in advance, even from major locations like the Microsoft campus.
Under most circumstances your best bet is to rent a car. Seattle airport has convenient car rental directly outside the airport (no need to take a "car rental shuttle"). I have made it to Redmond within 45 minutes of stepping off a plane, even including the time to rent a car.
If you absolutely don't want to drive yourself, you can take a private taxi, although once you get to Redmond, you'll be pretty much stuck wherever the taxi drops you because things are very, very far apart. There's also a shared-van service called SuperShuttle which is cheaper than a taxi but which will drop off a half dozen people at different locations so it takes longer.