Go to the main art gallery - then head to Transport (afternoon or weeknights only, fri/sat night the joint sucks) If you see a dark skinned woman with braided hair, then she is the beer Guru of the joint - have a chat to her if you can. If you think you know a lot about beer, she will prove to you that there is still a lot to learn. In fact, once you weed out the casuals and nuffers, there are at least three or four of the senior staff who are serious beer people. After a couple, head over to the other side of Federation square to the Ian Potter center for Australian art - right near that in Fed square is Beer Delux a newer beer bar/restaraunt.
After a couple, wander down swanston street to -
Cookie - once again, in the afternoon its great, at night too damn busy. A trick though.... eat there, IMHO the best Thai food in Melbourne, not the usual stuff, done with real flair and fantastic ingredients. The restaurant is separated from the maddening crowd, you will get a seat and a nice waiter will bring you your overpriced beer from the extensive list. Book a couple of days ahead or be prepared to get there either early (6:30ish) or late (post 9:00pm) for a sneaky no booking table. Also, if you happen to risk the place when its packed to the gills on fri-sat night; you can shifty your way into a table in the restaurant section after about 9-9:30 once it starts to empty out. You get a seat, aren't required to eat and might well still get table service.
For a more quirky drinking/eating experience, go upstairs in the same building as Cookie to the Toff in Town. French restaurant with a kinda cool train carriage style lay out. Serve lighter food for lunch, canapes etc early evening and dinner stuff later on, able to eat till 2:00am. Seriously girlfriend impressing territory!! But, last time I went there, you couldn't book. So if you want a booth or table you need to get there early and hunt one out. Give em a call though, it might have changed. They are sort of connected to Cookie, so there are at least a few good beers on tap, a great beer list and a frigging mind bleedingly huge winelist. The Age had a good review in the Epicure section, should be able to google it.
Upstairs in the same building again for the Rooftop bar - less impressive beer selection.. but there is a cinema. Check out Rooftop Cinema on Google... if cinema is not operating, though it should be over summer, the bar is pretty damn cool in and of itself and there will be Coopers and Creatures on tap (or there was last time I was there)
Nearest good beer joint to the museum is probably the Lambsgo bar in Collingwood (15min walk) - 3 Ravens bronze on handpump, usually something else decent on tap, and about 100 different great beers in the fridge. Limited "snacky" menu. The place itself is part of the attraction.. its set-up in an old terrace house and it kind of feels like you're back in your student days and at your mates place having a beer or two. On I think Friday and Sat nights the dark haired barmaid (forgotten her name, bad Thirsty!!) knows her beer and used to work at Buckleys micro brewery - quit because they didn't know squat about brewing!! Which if you have ever had a Buckleys beer, you would know proves she really does know her beer.
Quite near there is of course Brunswick st for great cafe culture and cool bars etc. Bimbo deluxe has good pizza and usually a decent beer or two.
As Fents said... The mountain goat brewery is a good option of Friday nights (every friday) and also on Wednesday nights. Fridays are busy, Wednesdays less so. But the place is big so its pretty good, and you get to ogle the brewery while you drink. Goat beers on tap and they sort of do snacks and average pizzas. BUT - - over the road from the Goat is the Royston hotel. A pub that concentrates on good beer. Imports in the fridge and Aussie beer on tap. There is CD and Stella to keep the rubes happy, but then there will also be 4 or 5 good micros on tap as well. Last time I was there they had Holbrook ESB on handpump. The place is also a great gastro pub. Really good food on the bar menu and excellent more up-market food in the dining room. Very reasonably priced and top notch quality.
There are more - but I'll let other people chip in as well. If you choose Goat one night.. don't be afraid to let us know. Just up the road from me and it would only be hospitable to buy an interstate AHBer a beer. In fact, same holds true for any of the venues I mentioned.
Have fun
Thirsty