I have ridden my bicycle from Seattle to Olympia, and vise versa numerous times, and it has struck me that there are probably a fair number of people who do not know how to do this, sooooooo… here are the three main ways I have gone. They are all very doable in a day, provided you leave early, bring enough food, and do not dawdle too much. If you are the sort to do this type of thing, they are all actually pretty doable in an afternoon, provided you keep a decent pace, and do not get lost. The shortest amount of time I have ever done this in is around four or five hours. That was time spent primarily in the drops, with a backpack, on a racy road bike. I went the Vashon route. For reference, the longest amount of time Olympia to Seattle has ever taken me was around eleven hours. This was with me riding a heavy, upright bike, with panniers. I took the Peninsula route in the dark.
#1: The Interurban Trail Route
This route is the flattest, most direct, and the only one that does not involve ferries. It also happens to be my least favorite, and as a result I have only taken it two or three time, including my attempted move from West Seattle to The Evergreen State College. The first 30 miles or so are fairly scenic and pretty, with lots of trees, and farm land, but after that ride get kind of gross and urban. I could not figure out how to embed this map, so here is a link.
You don't really have to worry about food, or other supplies on this one, because it is so urban, in fact you could probably bus to Seattle or Olympia pretty easily if you run into any sort of trouble along the way.
#2: The Vashon Island Route
This is my favorite route as of now. I like ferries, and I like Vashon Island, so this way is fantastic for me. I think this way might be a bit shorter than the other two, but it is definitely the hilliest. Multiple gears would probably be a good thing to have for this ride, although I have done it on a fixed gear, and you probably could too. Be advised that the ferry from Tacoma to Vashon will cost you around five dollars.
I am again having trouble with Google Maps, so here is another link. Google wants you to go through White Center, and up Airport Way after arriving in West Seattle, and will not allow me to conveniently tell you otherwise. Instead of doing what the map says, you should follow Fauntleroy Way SW North off the ferry, until you get to SW Avalon Way. Take Avalon, until SW Yancy Street, where you should take a right. Once at the bottom, find your way around the All Star Fitness, and take a left on Delridge Way SW. Get onto the side walk just before the street becomes the freeway. You will be on a bike path that will take you to East Marginal way. From there you can use the map.
#3: The Peninsula Route
This is a very pleasant route. I have only taken it once, but I quite enjoyed the ride. It is a bit hillier, and a bit longer than the Interurban route, but there is less traffic, more water, and better scenery in general than the first two. If you want an even less trafficked, and more scenic, albeit lengthier route, the find your way over to the 106, and ride along the water. until you get to Highway 3 in Belfair.
I hope this helps you ride your bicycle to or from Seattle sometime soon. Maybe you can come visit me. Have fun, and safe travels.