Seattle has a well-developed bicycle network consisting of several major urban and regional trails that allow for nearly road-free travel to all major neighborhoods and destinations in the Puget Sound area.
For the best overview of all Seattle regional trails see the 2015 Regional Trail Map from King County. It provides a nice summary for most of the major trails. The Seattle DOT also has several bicycling maps available.
Public transportation is available to extend the convenience of the trails; see here for information on loading bicycles on Sound Transit trains and buses. This page has information on loading bikes onto Metro buses.
View Burke-Gilman Trail in a larger map
2. South Lake Union Trails (Lake Cheshiahud Loop) and Ship Canal Trail
This pleasant loop route around Lake Union is almost entirely on bike trail except for the east side of the lake where there are low-volume roads along the shore. The Burke-Gilman trail makes up the northern portion of the loop between the University and Fremont bridges, and the western section has a paved trail along the water which connects the Fremont Bridge to Lake Union Park. The southern portion of the loop continues on the paved trail along the trolley tracks and Fairview where it ends. From there, signs direct you along the eastern shore to complete the loop.
At the Fremont Bridge, the Ship Canal Trail splits off the loop route and heads west along the south shore of the canal all the way under 15th and into Magnolia.
3. I-90 Trail (Mountains to Sound Greenway)
The most direct route to points east from Seattle. Connects to several major east side trails, and ultimately, central Washington State.
4. Seattle Waterfront Trails
Over 5 miles of waterfront trails can be pieced together from the Elliot Bay Trail and it's extensions.
View Seattle Waterfront Trails in a larger map
5. University of Washington Lakeside Trail
This short (1 to 1.5 miles depending on the route) trail runs along the ship canal and Lake Washington roughly parallel to the Burke-Gilman Trail, and through the Union Bay Natural Area on onto 45th where you can rejoin the Burke. The trail is wide and mostly hard-pack, you wont have any trouble on any type of bike.
6. Sodo Trail
Short, but useful trail that runs next to the light rail and allows you to bypass the busy roads and crappy pavement. Useful for getting to the stadiums from the south, easy access to the light rail. For general Sound Transit biking information, including light rail, start here.
View Sodo Trail in a larger map
7. West Seattle Trails
There are three major bike trails serving West Seattle, the first is short utilitarian trail that starts at the end of the East Marginal bike lane, crosses over the lower bridge into West Seattle, and intersects the start of the second trail, the Duwamish, before continuing on to a connection with Delridge Way. The Duwamish Trail follows the river south and connects to the South Park neighborhood and the Green River Trail. Delridge is the best route for getting to Fauntleroy and the higher elevations.
From Delridge, the trail continues across a confusing intersection where it ends at the third trail that leads along Puget Sound to Alki Beach. The Alki Beach Trail along the Sound is very popular (but with a separate path for wheels), has panoramic views of downtown and the Space Needle, and is one of the best trails to ride in the Seattle area.
North Seattle's primary north/south trail, connects Seattle to Everett along the Interurban Trolley route that ran during the early 1900s.
9. Interurban (South) and Green River Trails
The two main trails heading south from Seattle; the Interurban follows the former trolley route and is straight and direct, but mostly follows power lines through industrial Kent and Auburn. The Green River Trail follows the river and is a much more scenic ride, but a slower, more meandering one.
From the end of the Interurban Trail in Auburn, it's possible to get all the way to Orting, WA and beyond via the Sumner Link and Foothills Trails. The total mileage one-way to Orting from the Space Needle is 44 miles, and if you continue on to South Prairie, it's a little over 50 miles.
10. Chief Sealth Trail and Beacon Hill Greenway
A southern Seattle neighborhood greenway and trail with amazing views of Mt Rainier (and power lines)
Trails in the Meridian and Maple Valley areas, including the Cedar River Trail, Lake Wilderness Trail, and Soos Creek Trail.