The UW currently waives the non-resident tuition differential for out of state students who have graduate payroll appointments. While this has been a long standing policy, there is no guarantee that this will always be the case.
In order to protect yourself and the dept from some future bit of fiscal madness coming out of Olympia, you should take steps to establish Washington state residency, including obtaining a Washington driver's license or state ID, upon arrival. Please note that this will be particularly important if you end up choosing to do your thesis research in a lab based at the Fred Hutchinson Cancer Research Center, since tuition payments are processed differently than at the UW, and residency is very much an issue. Even if you are not based at FHCRC, there are sometimes tuition issues toward the end of your graduate studies. For example, occasionally a student will defend during the interim period between summer and autumn quarter and will then need to register for two credits during autumn quarter, even if they have wrapped everything up and will not be on UW payroll. In that case, the student would incur the much higher non-resident tuition rate.
You will need to apply for resident status in your second year by filling out the application posted here: http://www.washington.edu/students/reg/residency.html. Please disregard the intimidating verbiage and submit the application. International students do not need to bother with this, since the UW will not grant residency to international students.
The UW has recently made it more difficult to obtain residency. This guide, courtesy of the academic office at the Fred Hutchinson Cancer Research Center, will be very useful. Please direct any questions you may have to Brian Giebel (bgiebel [ a t ] uw.edu) rather than the FHCRC contact listed on the document.