Just seeing this, but depending on where you are in the city, these are my recommendations...
Avenue Pub. Period. Not a brewery, but the best beer and whisky bar in the Gulf South, IMO.
Courtyard Brewery. Great spot for hoppy beers. Scott (the owner) is from San Diego originally, so brews beers with that leaning. He also has a lot of guest taps for stuff that is different from his.
Wayward Owl. Just opening up, but open from Tuesdays through either Saturday or Sunday from 4-8 until their grand opening in a couple weeks.
Gordon Biersch. Even though it's a chain, they give the head brewer, Sonny, a LOT of leeway on one offs and rotating stuff. He has even improved the quality of the year round stuff compared to the last guy.
NOLA Brewing. Great space along the river. They also have a lot of experimental stuff in the tap room that you can't get anywhere else.
Second Line. Kind of hit and miss in my experience, but when they hit, they're good.
If you get a chance to drive across the Causeway to Mandeville, stop at Old Rail Brewing. It's a brewpub, so only on-premise and growlers, but Matt makes the best beer across the state, IMO.
Also beers to look for packaged (definitely hit up Stein's Deli for food and their packaged beer which is out the back door to the right) are Gnarly Barley and Parish.