Ballard (NW end of Seattle, generally) is a family-friendly area, but not the hot-spot for restaurants. Still, my BIL pays $2500 per month on a mortgage in a squeezy, modestly maintained home. Just warning you-- financially secure in the Midwest might not prepare you for the housing prices out West. But it's a good time to buy here.
I haven't been in Seattle for 7 years now. We sold our house at the height of the market. Seattle is a patchwork of dozens of nearly self-contained neighborhoods. Pretty much anywhere you live will be a short drive to groceries and bakeries and yoga studios There are coops in Greenlake (nice neighborhood--busy with through traffic and really expensive for the modest homes there, but near one of Seattle's favorite parks and the Zoo.), View Ridge, Fremont, Seward Park, all the same co-op and all those neighborhoods are nice.
North Seattle tends to be more "family-friendly" (View Ridge and Matthews Beach are popular), than areas like Capitol Hill (which have all the action). But everywhere in Seattle, except downtown, there are single family homes and yards, even in neighborhoods like Capitol Hill, and those are quite nice, though busy and potentially with higher crime. West Seattle is also nice for families, but the commute is worse because it is dependent on an overused bridge and amenities are fewer-- but some people like that. Plus, you are near Alki Beach and Lincoln Park--all nice places. Southwest Seattle is far more modest--Madrona is a popular one on the north-end of that, and Columbia City has it's own farmer's market-- but buses are slightly harder, and there is a patchwork of high-crime areas in that part of town. Really, you go from multi-million dollar home to the hotspot for crack in 2 blocks and I'm not exaggerating! (24th and Cherry used to be that intersection-- but also the home of Catfish Corner and a wonderful little coffeehouse once upon a time.) But that hasn't kept those areas from being well-loved.
Queen Anne is a nice compromise between proximity to downtown for ease of commute (and no bridge-bottlenecks to slow you down!) and "quiet" neighborhoods (traffic-wise, the roar of the city is pervasive). Not as close to the parks that are favorites of families (Matthews Beach, Lincoln, Ravenna, Golden Gardens. Seward Park), but still nice and short hop to downtown, Fremont and Phinney. Phinney is great as well--right next to Greenlake without the vicious prices or through-traffic. Lenora-- Walingford--lord the list is endless......
School districts-- Seattle is one district, but individual schools have their own "flavor". I'm afraid because we homeschool now and I left Seattle when dd1 was 1yo, I can't guide you here. My friends in View Ridge take their kids to "AE-2" I think it's called, a public, alternative-ish school. There is a Waldorf school in the Meadowbrook neighborhood (my old stomping grounds). Elementary schools are not based solely on neighborhoods these days, unfortunately, but that means you can base where you live separately from the schools, except for commuting concerns.
I lean towards North Seattle simply because know it best, not because there aren't good areas South.
So, you'll want to decide how busy and bustling you want your neighborhood to be, first of all, and how urban-or-suburban you like the flavor of the neighborhood to be. You aren't going to be too far from anything, relatively. Know that cross-town traffic is slow at the best of times-- the entire city is oriented North-South. I think commute-times are going to be a bigger consideration from everyday errands and park trips. You'll not be too hard put to get to those in a reasonable amount of time.