Hi, I'll pitch in. The only bridge that doesn't allow bikes is the 520 bridge across lake washington.
For rides in the city from Woodland park, 2 suggestions, ride west until you hit Puget Sound, then ride south across the ballard locks, then out to West Seattle for views of the city and the sound. Or, ride south until you hit Lake union, head east along the lake on the Burke-Gilman trail. ride out to the Red Hook Brewery, drink, then repeat the ride home.
Check out
www.cascade.org for some rides you can join up with, and the city of Seattle has some great bicycling maps you can pick up for free at most bike shops. Closest shop from Woodland park is Gregg's Cycle on Green Lake.