I moved away from Birmingham back in the mid 2000s and then came back several years ago. The city has come a long way, but it's still relatively backward, especially as compared to Atlanta, Nashville, and even Huntsville.
Avondale, just east of downtown, has seen a huge resurgence. It was crime-ridden, but has come back to the point where I don't even think hipsters are willing to live there anymore. Avondale and Cahaba breweries are both in this area. Saw's Soul Kitchen, especially the smoked wings, and Melt are restaurants worth checking out, too.
Lakeview is another area, kind of on the southeast margin of downtown, with a mix of bars and restaurants. Moe's BBQ in this area is quality. The B'ham location of FIVE is here too, although the Tuscaloosa location is probably better known.
Central downtown is where you'll find Regions Field and Railroad Park, which are the city's attempts to spur revitalization. Good People Brewery is near the ballpark, but the bar scene will be slightly north of there - in the Morris and 1st Ave. North area. Paramount on the northside has fantastic food and an arcade, for anyone pining for the 1980s.
Five points south is still a thing, but not really for the bar scene anymore. There are still a few quality restaurants though, including Highlands, Frank Stitt's restaurant of the year winner, and its sister restaurant, Chez FonFon.
Lastly, there's the civic center area. There are a number of restaurants here, but they seem pretty cookie cutter to me. Not Olive Garden or Applebee's level, just not anything unique or special. There is Top Golf in this area now, if that's your thing.