I live in Morningside on the south side of town. It's the neighborhood centered around the intersection of Morningside Road / Balcarres St / Comiston Road / Cluny Gardens. It's a safe friendly place, and there's great local shops and pubs, plenty of nice old stone tenement buildings with flats for rent and lots of buses straight in to town. Nice parks, hills and green spaces as well for a wee walk. Oh, and Maisie the cat lived here
The Royal Edinburgh Hospital and Astley Ainslie Hospital are both in the neighborhood, or the 38 bus will take you direct to the Royal Infirmary one way and the Western General Hospital the other way. So excellent access to hospitals.
Teaching I'm a bit less sure. The local schools on the south side of Edinburgh are considered very good but schools are obviously scattered all round the city so probably depends where you get a job.