As a real estate appraiser for 27 years my experience is the safe neighborhoods are abundant. Check out West Knoxville as it is the most developed and offers more shopping, restaurants and medical centers. Hardin Valley has a newer elementary school. Housing prices are higher, but still very affordable. Hardin Valley as well as many others places still have a country style feel with some open space or woodlands. There are many surrounding areas with nice subdivisions that are like a quiet oasis removed from the hustle and bustle of city life, yet only a few minutes to all the amenities a busy mom needs to raise a family.
If you want real country life in a low density residential area, whether within a nice subdivision or on a small acreage tract, they are abundant and safe, and within 30 minutes to all the things a busy mom needs to raise a family. I regularly travel back country roads and have never been harassed or felt endangered in any way. Very quaint really because on the back roads people will actually wave to you as you pass each other in your cars, or from their front porch. Yes, seems a little silly but if you ever need help, for any reason, people will stop and go to almost any length to help you out. I regularly need to stop at a strangers house and ask a question or two. I have always, always been greeted as a friend. A few people are wealthy but most are retirees or typical families that are intact and on an average income.