Before we moved to Durham (outside of the city, but in north Durham County), I spent a lot of time on the Durham Crime Mapper
Durham Crime Mapper. It's a good way to see what kind of crimes have happened in what neighborhoods in Durham (including city & county). The comparative crime site is good too, as it can be hard to look at Durham's great Crime Mapper without knowing how crime stats compare to say, Carrboro.
One of my favorite things about Durham is the weekly Farmer's Market, which is quite close to the Durham 'hoods that have been recommended.
Regarding roads, beware that there are "business" and "bypass" versions of the same route number. I know this is true for 15-501 and 70 (in Hillsborough), and it confuses me just about every time.
Durham is a great place, no question! Good luck!