Anywhere within loop 12 will have acceptable public bus transportation. Outside of loop 12, buses are less practical, only because the territory to be covered by the network of bus routes is larger... as in PI times the square of the radius...
DART rail operates on a 10 to 20 minute schedule. It currently has about 45 miles of rail and about 35 stations. Virtually anything you'd want to buy or entertain yourself with is adjacent to one of those stations, as are many residential units. It's true that there is not a rail station on every street corner, but the areas close to the rail stations comprise a city within a city that will fill every need you have.
For a rundown of the DART rail and bus system, check out
DART.org - Dallas Area Rapid Transit. People who claim it's "pretty bad" are never able to exactly define their opinions.
As far as neighborhoods that are easily livable on foot, you'd want to be in the downtown or Uptown areas, or Oak Lawn. Areas adjacent to shopping centers are usually good for walking to shopping, and there are several "New Urbanism" centers, such as Addison Circle, Legacy Town Center, Mockingbird Station, Watters Creek, etc, that have high density and walkable centers.