Here is a write up I did a few months ago for another forum, let me know if you have any specific questions or would be interested in an IP trade...
Breweries:
Yazoo - it's the most consistent and oldest, and usually putting out some solid stuff. They are currently teamed up with Brandon Jones who's a phenomenal homebrewer and the author of Embrace the Funk blog. They release very limited beers in their Embrace the Funk series. Some are funky and some are sour, but only one is bottled so far and others go on tap sporadically. Some people get upset that they only see Yazoo's regular lineup on shelves and unfortunately that is all they bottle. There have been a few one off things released for events in the past 3 years I've lived here, but otherwise it's just the regulars on the low grav side and bombers of Sue on the high grav (liquor store side)
Jackalope - They are indeed about a 3 minute walk from Yazoo's front door... If you want some extra ticks on Untappd, might as well stop in; otherwise they just aren't worth it, IMO. Very inconsistent and subpar beer coming from them
Fat Bottom - Pretty awesome taproom and decent food, but the beer isn't quite there yet. It's not terrible, but definitely nothing to make you want to go back. You can usually find one or two taps at almost any bar that serves some sort of craft in town
Cool Springs Brewery - about 15 miles south of Nashville this place is really hit or miss. They do have I think 4 regulars on tap, which are decent and drinkable beers. But, there are usually pretty unique ingredients going into the 2-4 rotating taps they have on at any given time and sometimes they are awesome and sometimes I can't finish the pint.
Turtle Anarchy - Alright, these are the guys to hit when you visit if at all possible. They are located down in Franklin (not far from Cool Springs Brewery), which can be a 20-30min drive depending where you are in Nashville, but they are worth it. They have a solid ass lineup (one of which is an incredible stout) and they also have an experimental stout series called Fifty Shades of Black - playing with anything from weird ingredients to barrels. Check their facebook or call ahead to see what's on cuz it's never the same and sometimes they don't have any from the series on tap (though more rarely now they've been doing it awhile).
Black Abbey - this is the second newest brewery in town. Not amazing, but IMO worth hitting as it's a short drive from downtown. Kinda a cool taproom that is a sectioned off part of the brewery and some pretty killer jalapeno rolls as a snack.
Tennessee Brew Works - The newest kids on the block, these dudes put some damn money into their place. A huge two story taproom with some solid beer. They appear to have 5 regulars now and a couple test batches so far and they are going to be up there with the best in town if they keep it up...
Blackstone and Boscos- they are the brewpubs that have been around for a long time. Boscos has the better food and atmosphere, but Blackstone has the better beer - and they are starting to get into barrel aging. They have a bourbon barrel imperial milk stout on tap now and a BA RIS blend coming to bottles soon!
Shops:
Midtown Wine and Spirits- They always have a big selection and get the limited stuff in decent quantity. Also one of the only places that allows you to make a mix 6er of almost anything on the low grav side (under 6.3%)
Frugals- not quite the selection or the quality of stuff as Midtown, but they are close to Yazoo/Jackalope and convenient to hit
Craft Brewed Nashville- the new place in town. They are a store, bar, and homebrew shop. They are quick to get into the scene and from what I've seen now get the highest quantity of limited stuff. The taps they have (about 20) rotate very frequently and offer some local stuff that doesn't go elsewhere. High grav store and low grav store/bar are right next to each other and just a 5 minute drive away from Yazoo. Service (especially on the high grav side) is second to none in the local beer scene.