I've lived in Atlanta my whole life, so I'm not sure how it compares to other cities of it's size. There are definitely places that are "safer" than others, but every neighborhood has it's "hot spots" where crime is more likely. I've never heard of pick-pocketing, but plenty of phone-snatching (walking around with headphones on and your phone in your hand) in places where there is an easy escape route (transit stations, for example).
Looking at crime statistics is always scary, so I would avoid those and try to ask around on local Facebook groups to find neighborhoods that fit your lifestyle and commute. I've lived all over the east side of Atlanta (East Atlanta Village, Glenwood Park, Inman Park, Little Five Points) and currently live in Reynoldstown, and I think it's fantastic here.
Feel free to reach out about particular neighborhoods that you have your eye on, and I'll give you my impression of safety and general liveable-ness.