Where would you be starting in Seattle? And what are your riding capabilities, interests, etc?
Ceedbux's, Headloss' and Crimsonsong's recommendations are pretty good. The Burke-Gilman trail is a local treasure, but also very well-used. It traverses some of the city's nicer parts, including the canal between Puget Sound and Lake Washington, the UW campus (current detours go right into the campus), and really nice waterfront neighborhoods on Lake Washington. If you're up for more miles, it easily connects with the Lake Sammamish trail that goes through rural areas and passes some excellent wineries and breweries before finishing in Redmond, not far from the Microsoft campus. There's an excellent bike path across the I-90 bridge on the Lake, but it can be a bit tricky to find it's starting points. If you'd been here last weekend you could have seen, or been one of the many naked bicyclists in the Fremont (neighborhood near the SW end of the trail) Summer Solstice parade.
The 10-mile loop around Mercer Island (accessed by the I-90 bike path) is pretty, fun riding, and you'll go by Paul Allen's (one of Microsoft's founders, owner of Seahawks and Trailblazers, among other things) house or at least the gate into it along the way. VERY nice neighborhood!
if you ride to the Fauntleroy ferry in West Seattle, then Vashon Island, you'll sample one of our more typical urban areas, have the option to ride along one of our nicest beaches (Alki, very busy on sunny days) and my home area. The ferry is nice and free for cyclists eastbound. Vashon is a lovely rural area, but be prepared for hills. I spent most of this winter's weekends there, and there's lots of climbing per mile there, especially the 1st mile climbing about 400' from the ferry dock. Any of the roads on VI are great riding. Pick up an island map at a real estate office along the way. The island is 14 miles long (about the same size as Manhattan, with 1/150 the population), and the town of Vashon has a several nice restaurants, plus others along the road that runs through the center of the island. There's one winery (Palouse) with daily tastings just past the climb from the ferry, and being Seattle, plenty of places to get good coffee.