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.
Another library option is the King County Library System. Seattle is in King County, so it's not clear why there are two entirely different library systems, but there are. The KCLS locations are not in Seattle proper. However, it's often worth it when you're looking for a hard-to-find book to check out their system as well--they seem to have more of a selection in many areas. Holds can be picked up from any KCLS location. You can get a KCLS library card if you live in the "King County Library System Service Area" or in Seattle, Renton or Enumclaw. KCLS generally has all the amenities of SPL--except for the beautiful Central Library.