How important is proximity to train vs bus vs driving?
On the closer to the city side, you might like Ballard, Phinney Ridge, Tangletown, Wallingford, Ravenna, Greenwood, Maple Leaf. All will have some areas with quiet residential streets which are walkable to some neighborhood commercial area. Of those, Ravenna/Tangletown/Wallingford have probably better train/downtown access - the rest will still have bus routes towards downtown though the bus network isn't totally comprehensive.
Some areas further north could be worth looking at though I'm not as familiar with them - I would guess they are less expensive with potential for more space, though potentially less walkable neighborhoods. The light rail is opening an extension all the way to Lynwood in just a couple months, so it could be interesting to look at Shoreline or Mountlake Terrace if you don't mind being a bit farther away but still on a train line. I would guess the vibes are a bit more strip-mall-y the further out you get
West Seattle is known for being pretty family friendly, you pretty much need to drive if you want to get to elsewhere in the city but it's beautiful there and it still has walkable areas.
I've heard good things about the south neighborhoods like Beacon Hill, and those do also have light rail access which is good! But I'm not as familiar with what the neighborhoods are like
Fremont's nice too! No train access but it'll probably have good bus routes to SLU since it's right along the bridge. Similar to the others I mentioned it should have some quiet residential streets that are still walkable to commercial areas. I would guess it'll be on the expensive side for buying depending on how much space you're planning for family-wise. It's a nice area!
In that vein similar to Fremont, Queen Anne could be a good option too. I also suspect on the expensive side because it's fairly central. Also no train but there should be good bus access to SLU and downtown
I think the slightly more northern areas like Ballard, phinney, Ravenna, tangletown, maple leaf will have more big green parks close by and potentially larger lot sizes. Fremont, Queen Anne are closer to downtown and SLU so probably quicker commute and should still have options for quiet residential spots, and I would guess higher prices for buying with smaller lots