Moved from Seattle 6 years ago, but the North End in general is still really family friendly compared to the rest of the city. Fremont can feel distinctly urban, depending on which street you live on, a thumbs-down for me. In general, staying close to a neighborhood hub will make a bigger difference on walking/biking for a family than any one particular neighborhood. All neighborhood hubs will have decent buses. For bike friendliness, do you mean with your kids? Or on your own?
Greenlake is fun but prohibitively expensive these days. Lake City is cheaper, but without the charm and pedestrian-friendly feel, yet is still supremely walkable and has unbeatable commuter buses along Lake City Way (Bothell Highway) if necessary. Downtown 10 minutes, no joke! In the AM, you'd never have to wait more than 7-10 minutes for another bus. Much less expensive than other, trendier neighborhoods. You have a large library, several grocery stores, Value Village, quiet, shady, kid-friendly streets, etc. *Zero* charm, but you can visit "charm" and still have money to spare for the extras!
We lived just outside Lake City-proper in the Meadowbrook neighborhood. Buses were still great along Sand Point Way (to UW, Magnuson Park, Matthews Beach, U Village), the Burke-Gilman trail was nearby (but not without going up and down a hill) and streets were quiet. Not as walkable to grocery stores, etc.
Just uphill from Meadowbrook is the Wedgwood neighborhood, still far less expensive than Greenlake and a favorite neighborhood of our friends.
I still like the west end of Ballard/Sunset Hill. Decent buses, the Ballard Locks, and the largest consignment store in the city (Mom'n'Me). Good busses to Golden Gardens, a favorite of mine, and good walking once you are in downtown Ballard.
Phinney/Greenwood is the "new Greenlake" (read: all the cheap houses in Greenlake have been taken and this is the next best thing.) Still near the Greenlake amenities but without all the busyness, and *walk* to the zoo/Woodland Park (annual pass for a great deal). I would much prefer Greenwood to Greenlake, myself (if I even had the money to think about living there. Seriously--when we made the decision to put our house on the market and move close to our property outside Olympia, we knew we would *never* be able to afford living in Seattle again! The house dh purchased for 90,000 was sold for 285,000, a bargain at the height of the market.)
I pretty much found good support wherever I went, but I was still just with an infant at the time and moved shortly after.
Hopefully you'll get some good responses from parents still living there.