Well, “fast access” via public transportation is always something of a dubious proposition around here. For the northern tip of the lake, you might want to look at neighborhoods like Shoreline, Lake Forest Park, Lake City (maybe–the yards are going to start getting smaller here), Kenmore, Bothell (maybe–you’re starting to get into “Microsoft bedroom community” territory here), or Brier (maybe–would probably require more driving).
I’m not familiar with the southern tip of the lake, so I can’t give good advice for that area.
I grew up in Rainer Valley, about half a mile south of Columbia City. I returned to the valley to mentor a friend of mine’s kid through little league. When I grew up, the valley was mostly lower middle class white with very few of us minorities. It has become quite diverse now. Columbia City is more or less a four block long stretch of urban renewal.
The little group of business to the north at Genesee street, and Hillman city, about half a mile south of Columbia city show little signs of improvement.
House prices are lower in part because the mint route to downtown is still Rainier Avenue and has always been a tough half hour commute. Mount Baker ridge above Lake Washington, any place with a view of the lake, and the Seward Park areas are still desirable neighborhoods with Caucasian enclaves. The ridge to the east of Columbia city has what I always considered nice middle class houses. To the west, the houses are less desirable and the income levels are lower. You can see this in Columbia City itself as the good restaurants, the bakery and the Starbucks are all on the East side of the avenue.
I would not recommend the area if you are raising children, unless you need the more affordable housing. The problem is weith the Middle School and High School students. You will be exposed to the anti-intellectual influence of the punks (like I say, I grew up in the area and they certainly had a negative influence on me).
Living in South Seattle would be similar to living in the villages of Alaska. Although the people are quite friendly, you will always be an outsider.