The Seattle area offers many local car-free bike trails, most notably the Burke Gilman Trail, as well as wide roadways and a fast-growing number of dedicated bike lanes. The Burke Gilman Trail is 27 miles long and the terrain stays fairly flat, so it is suitable for families as well as more serious cyclists. The trails run from Golden Gardens Park in Ballard all the way to Redmond's Marymoor Park via the Sammamish River Trail. It has many access points, making it easy to ride just a few miles or all 27 with many stops along the way at beaches, breweries, parks and the University of Washington campus.
A great resource for cyclists is the Cascade Bicycling Club, a non-profit organization with over 16,000 members (in fact, Cascade is the largest cycling club in the country). The club organizes daily rides for all skill levels as well as big events like the Seattle to Portland, Oregon (STP) ride every July, and is a leader in bicycle advocacy and education. Check out http://cascade.org for more info.
The Seattle Times also published a brief article for family-friendly bike rides that featured one of our favorites - "Bicycle Sunday" is a very flat, 8.5 mile ride along Lake Washington via the lovely Olmsted-designed roadway. The ride has been taking place one Sunday of every month from May to September for the past 50 years.
The roadway is closed to car traffic on this day, beginning just south of Mt. Baker Beach and ending at Seward Park, where you'll find a play area, beach and perfect picnic spot. While this ride is appropriate for all levels of cyclists, it is particularly inviting to kids with training wheels and parents pulling bike trailers. The route is a popular ride in general, but is recommended for children only when the road is closed to car traffic during "Bicycle Sunday" events.