For your Dallas to Chattanooga leg... (or at least the AL/GA/TN portion)

In Birmingham Trim Tab would be my first choice, Good People is solid as well. Hop City is a great bottle shop with a bar, the J. Clyde is a great beer bar.

You could route through Atlanta and take 75 up to Chattanooga. In Atlanta I'd recommend Atlanta Brewing Co, Monday Night (go to the Garage location), Orpheus, and Three Taverns - several others as well but those are my tops. Also, Wrecking Bar Brewpub is fantastic. Brick Store Pub is excellent, but it's not going to give you the best taste of Georgia, if that's what you're looking for.

In Chattanooga Heaven and Ale is my first choice, the OddStory, Hutton & Smith is decent and I thought their taproom had a great/ friendly vibe.