Well....
As NRA4Life said, you have already missed the boat for this year as far as special permit applications go.
As a military person stationed here, you will be able to buy resident price tags, which is nice. Non-resident stuff costs 10 times what a resident pays, and I am not sure that hunting in Washington is good enough to justify paying $700+ for a deer/elk/beer/cougar combo.
First thing, swing into a sporting goods store and pick up the Washington State Big Game Rules. Then, go find a layer...
Just kidding...the regs here this year are about 103 pages long, but about half of that is the written description for the GMU's(Game Managment Area's).
Next, if you were born after 1971, you must attened a Hunters Education Class.
This is a link to the Official Washington State Website for places where the training is conducted.
http://wdfw.wa.gov/enf/huntered/basic_class.htm
I don't believe it is a 100% complete list, but it is a good place to start. It also wouldn't hurt to ask the folks at a local sporting goods store. If you are at Ft. Lewis, I recommend either Sportco in Fife, or the new Sportsman's Warehouse in Lacey.
Let's see...from there, it's a matter of finding some land. I'm afraid I can't help much with that...I have hunted public lands between Ellensburg and Yakima...Manastash, Umtanum, and Bethel GMU's. Opening day of Modern deer and elk are enough to make you want to dig a fox hole on public land anymore.
As for your gun..there should be nothing in Washington you can't hunt(other than birds) with a savage .30-06.
If you have any other questions, just ask...
greg