I have lived in Omaha since the fall of 1974. Most of the shootings are gang members targeting each other. Strangers just driving through are rarely bothered. The mantra about not doing stupid things with stupid people in stupid places at a stupid time of day holds true. A few years ago, a respected young business woman was murdered in her car late one evening. She had picked the wrong drug dealer.
The Bloods and Crips moved into North Omaha about 25 years ago. Instead of jumping on them immediately, the city's leaders spent the next few years denying that there was a gang problem. South Omaha has traditionally been the first home for immigrants. Fifty years ago, they came from central Europe. Now they come from Latin America. Their gangs have come, too. There are about 3,000 gang members and associates in Omaha.
North Omaha and South Omaha are not good places to live. It's too simlistic to define either area by a set of bordering streets. Another post mentioned Dundee and Benson both of which lie in North Omaha. Dundee contains older mansions I can't afford. It also has Brownell-Talbot, an exclusive private school for rich people's kids. Benson still has problems. Downtown, which used to be bad, has been refurbished. Little attention has been paid to South Omaha. A gross oversimplification, which excludes many good neighborhoods, would be to stay west of 72nd Street.
The various bedroom suburbs that are part of metropolitan Omaha are good places. If you are willing to drive farther, there are separate towns like Blair, Plattsmouth, Springfield and Ashland. You can also live on the east side of the Missouri River in Iowa. Just avoid the seedier parts of Council Bluffs.
As mentioned, the City of Omaha is fairly anti-gun. It helps to get a concealed carry permit since state law exempts permit holders from Omaha's silliness. One thing to remember is that, in Nebraska, it is illegal to carry inside a facility posted with a No Guns sign. There is no standard format for the sign. In Iowa, the worst they can do is order you to leave and cite you for trespassing if you refuse.
The Douglas County prosecutor's office has always been staunchly in favor of self defense. They do not prosecute victims who fight back. Neither have I heard of lawsuits by the bad guys or their surviving relatives.