In the UK and many other countries, the license is transmissions dependent, meaning that if you take the test with an automatic, you are only licensed to drive an automatic. If you take the test with an manual, however, you are licensed to drive both manuals and automatics. That is what mattythedog is saying.
In the US, however, that rule doesn't apply. Licenses don't make that distinction.
With that said, I would recommend you learn to drive an automatic first, then learn to drive a manual in a closed course until you are comfortable. I make this recommendation because you should get used to controlling a vehicle on the open road with other drivers first. The less stuff you have to worry about, and the more you can focus on just driving, the better the first couple of lessons will be. Once you've got a handle on driving a car, then you can learn to drive a manual. Especially in Seattle where there are tons of steep hills, you will have to learn good clutch control to start properly on the hills. I think it's better for you to not have to worry about that when you first start driving. Once you've got driving itself down, then you can start learning that and worrying about that.
If you were in a country where the license changes depending on the test transmission type, then by all means, learn a manual, but since it doesn't matter in the US, you can learn manual later, on your own time, after you are comfortable with driving. You should learn to drive a manual eventually (everyone should know how to drive one), but I don't think you need to do that in your first ever driving lesson. You'll have enough to worry about as it is. Plus, if you do take a driving school, I doubt that any of them will even have a manual car for you to learn on.
Learning to drive itself can be stressful enough, no need to add the starting-on-a-steep-hill-and-possibly-rolling-backwards-into-the-car-behind-you-so-you-either-give-it-too-much-gas-and-launch-like-some-lunatic-or-not-enough-gas-and-stall-and-roll-backwards-into-the-guy-behind-you stress anyways.
