If you are looking for a public library in Seattle, a good starting point would be the Seattle Public Library website, which has a list of locations, events, and more.
The SPL is not only great for research and borrowing books. They also have an extensive movie collection, both in VHS and DVD. With a library card and using their website you can reserve ("put on hold") any book or movie and they'll send it to your closest branch.
The Central Library is the flagship location and worth visiting, even if you don't plan on doing any actual work there.