I live in Burlington and here is my list of Dos:
Burlington
Magic Hat: You just have to, nothing impressive, but they have one of the best merch/tasting setups I've seen.
Flat Bread: Go for dinner, but get there early because there will be a wait. They always have good beer on tap, as well as their house brew zero gravity.
FarmHouse: The farmhouse doesn't brew anything, but they have the best selection of taps in town. Always something interesting and always worth it.
Das Beir Haus: This place is a bit pricey however they have the freshest pour of Aventinus I've ever had. I don't know how they get it, but it is generally fresh even verging on green.
Brattleboro
Mcneils: Solid brews of the english american hybrid. I haven't made it to the bar myself and I've heard rumors of dirty taps, but the beer is delicious.
Middlebury Area
Lawson's Liquids: This is a nano and you'll have to get a hold of the guy if you wanna stop by. His beer is generally in high demand and very good, chances are you won't be able to get a hold of any of it.
Waterbury/Stowe
Trapp Family Lodge: Yes...the sound of music. Vermont is where the family settled and opened a lodge/inn. They have recently (last two years?) started brewing authentic German beers. If you like things German they are delicious.
Alchemist: These guys have a very nice selection of house brews. All very solid and tasty. I'm partial to the holy cow.
Black Back:This is my new favorite bar. It is right across the street from the Alchemist and you could very well miss it. It is about an 8 seat bar, probably 12 person capacity for the room, however there is always a stellar selection on tap. He will always have something interesting from Hill Farmstead too (a new VT rockstar brewer).
Greensborough Bend
Hill Farmstead: If you really want to make a pilgrimage there is hill farmstead, which is in the middle of nowhere. This guy just opened his doors this summer and is brewing some very good beers. He started with IPAs, but has recently been doing a series of barrel aged sours, spontaneous fermentation, saisons and others. He also makes an amazing porter called twilight of idles.
There are many other selections and opinion, but this is what I'd do if I moved away and was coming back for a visit.