Check out the Edgewood Park\Southcliff neighborhood. Very nice homes, mature landscaping, well kept. Five to ten minutes from stores etc. The Felida area (not on the map,but it's north of the Northwest neighborhood) is nice too, but some areas are still fairly new and kind of sterile looking. The Lincoln area is historic, but they are smaller homes built by the Army back during WWII. If your looking for high budget homes, the Columbia Way Old Evergreen area along the river is nice.
Other nice areas are Evergreen Highlands, Du Bois Park, Bella Vista and the surrounding neighborhoods. The ones I'd avoid are the Rose Village, Fourth Plain Village, Harney Heights, Central Park and Fruit Valley. They all have their charm, but also have a lot of crime.
Here's the map.