Check out Eastwood.
http://walkeastwood.org
As safe as any area in the city, and its main business district, James Street, has an independent movie theater, two independent book stores, several small restaurants, a post office, ice cream, convenience stores, banks, gas stations, a cafe that roasts all its own coffee, some of the best pizza and wings (okay, the best pizza is Gianni's, still in Eastwood but down on Burnet near Teall).... the list goes on. Within 1 mile of two huge grocery stores and very near an entire shopping plaza. Wonderful parks, you can golf in the summer and cross-country ski in Sunnycrest Park. And several bus lines. Your rental budget will get you the very best in the area with some to spare. Each year this neighborhood has more and more young couples, with or without young children.