Seattle, WA to Olympia, WA via Tacoma or via West Seattle, Vashon and Tacoma
These are just notes...need to write up a detailed description but hopefully these will help in your planning!
See http://www.kingcounty.gov/recreation/parks/trails/regionaltrailssystem.aspx for a map of the bike trails in King County.
Option 1 - Inland
South from Seattle basically the nicest route is to follow Lake Washington shoreline south and then connect to the Green River trail which will bring you most of the way to Tacoma then onto Olympia.
Option 2 - Via Vashon Island
A more scenic and rural option is head to West Seattle, take the ferry from Fauntleroy to the North end of Vashon Island which has great cycling. You can take a ferry at the South end of the island to near Tacoma and miss much of the Seattle suburbs.
South from Tacoma I've only bike once and it was a long time ago. If you head to the coast it's a nice route but then Portland will be way out of the way if you want to head there before heading to the Oregon Coast south.