I live in Seattle too- the Waterfront Outdoor Center, Aquasports and both the Washington Kayak Club (WKC) and University Kayak Club (UKC, for UW affiliated folks) have pool sessions, instruction, and regular *safe* trips where you can learn the ropes with lots of back-up. Like the folks here, never white water alone. The UKC has a bunch of boats to use for free, plus warm gear , helmets and paddles so that you can try out what works best for you- Jackson Fun, Burns, Diesel, etc. With the WKC you need to get your own stuff.
Paddling up here means COLD water almost year round. At some point everyone gets a dry suit, but wait on that until you figure out if you are really going to stick with the sport. First order of business is to learn to roll. I can't stress this enough. The video called "The Kayak Roll, Learn, Tune, Teach" by Kent Ford is absolutely fantastic. It is much better to do this in a poool session than Lake Washington and Puget Sound is way too cold for learning in.
Again to emphasize- never boat alone, always wear a pfd and a helmet. I even do that when I'm doing roll practice out in the Lake.
See you on the river