The average SAT score for admitted students at University of Washington is currently an 1815, but that's for the old version of the test that was scored on a 2400-point scale. Since you'll be taking the new SAT out of 1600 points, you should aim for around a 1250 to have a solid chance of admission (based on this calculator).
As for study tips, I think your first step should be to take a practice test and see how you do. The College Board has four sample practice tests for the new SAT available here. Time yourself to the specifications of the real test so you can see if you have any time management issues. Then, score the practice test and see how far you have to go to reach your goal.
Consult this article for an estimate of how many hours you should devote to studying. You may not need as much time as other students to see big point improvements because you've never seen the test before, so you'll gain a lot of ground as you get used to the format.
The most effective way to study is to go through your mistakes, figure out which types of questions are hardest for you, and do additional practice along those lines. When you feel that you've made significant progress, take a second practice test and reevaluate your score level. You can also consult this article for an overview of the best way to study for each section of the new SAT!