The whole town is as bike friendly as it gets, but as others have suggested there are ample bike route guides for a clearer picture of where to go. More suggestions:

Ballard: Sonic Boom record store is one of the best in town. (They have a smaller one in Fremont.) Can ride out to Golden Gardens from here to check out the shore along Puget Sound. The Ballard Locks are also cool if you're into boats, fish ladders, gardens, and the engineering behind displacing large quantities of water.

Fremont: Lots of neo hipster shops. Conveniently located along B-G trail and other bike lane convergences.

U-District: More shops and record stores. Said Recycles Cycles always has a fixie or two locked up out front. Also along the B-G.

Lower Queen Anne: More shops and record stores. Home of Counterbalance bike shop, home away from home of many a messenger and fixie steed.

Lake Washington Blvd: If you want a super mellow, scenic road ride, this is lovely. Goes along the west side of Lake Washington. About 20m if you start in the UDistrict.

Burke-Gilman trail: If you start in Ballard and go all the way around to the Marymoore Velodrome in Redmond, it's about 60 miles of near flatness round trip. Lots of scenery and a good place to be mellow, but on a nice day watch out for the peds/dogs/rollerbladers/freds/hellonwheelsroadies, etc. [165], Habitus, friend-of-Habitus, and myself did the B-G this weekend out to the Redhook brewery in Woodinville for lunch and brews. A very fun trip, even funner when you're able to draft behind people against the returning headwind.

Glad to hear that you're bringing your bike out! You'll be happy you did.