I've heard enough whitewater kayakers tell me that when they first came to Seattle they had a hard time finding paddling buddies and generally getting started, so here are some good resources:
You can post on these pages and find people doing runs in your ability range.
Washington Kayak Club Facebook Group - Probably the easiest way to reach people belonging to WKC, and typically they will be running beginner/intermediate trips. I'm not sure if you need to be a member to be part of the FB group.
Robe Canyon Facebook Group - for the Class V Robe Canyon run (South Fork of the Stillaguamish River) about an hour from Seattle.
Ernie's Canyon Facebook Group - If Robe Canyon isn't hard enough for you, this is the page for the sievey and technical Class V on the North Fork of the Snoqualmie River.
ProfessorPaddle.com (PP) used to be the core whitewater kayaking forum and river resource for the general Seattle area. The social aspect is way toned down now because of Facebook, but you can still find some discussions and also descriptions of pretty much every river in Washington state (and the U.S.) under the 'Rivers' section.
* Just keep in mind that the gauge reading for the river on Professor Paddle will probably be outdated, click on the NWRFC link and check the gauge date to make sure you're looking at the correct level.
If You're Going to Oregon
The Portland area has its own version of ProfessorPaddle, called PDXKayaker.org. I've never posted on there but you could try it if you're heading that way. The one time I was down in that area by myself I missed up on meeting my Seattle area friends for a Canyon Creek of the Lewis run and two separate groups of PDX kayakers invited me to tag along with them - I ended up paddling with a huge bunch of people and it was a great day. I've always thought Seattle boaters were really open and friendly to new paddlers, but if anything the PDX boaters were even more actively welcoming - I was super impressed.
If You're Desperate :)
You can email me via the 'Contact' tab of this blog and Mike and I can try to put you in touch with paddlers who are doing runs you would like to get on, or we might be able to run something with you depending on when it is. I've met some nice people from Texas and Colorado that way (sorry CO peeps for the swims on Boulder Drop :). Just understand that we are usually pretty busy with work so it's a bit of a crapshoot.