Rock Bottom is real touristy, right in the middle of the theater district. It feels like a Fridays with beer.
Publick House is a T ride away on the C-line. Amazing beer bar.
As for a bottle shop Publick House Provisions is right next door. They are a little steep but have great hard to find stuff.
My favorite place is Penguin Pizza in Brigham Circle (Green line E train). It's a good pizza place with a great beer list. 40+ taps and hundreds of bottles.
Beer Works is right next to the Garden and is great for pre-game. It's a brew pub.
If you get there real early, Pizzeria Regina is amazing. It's very near the garden, in the North End. Their sausage, Bell peppers and mushroom pizza is my single favorite pizza.
Bukowski's Tavern is my second favorite. If they have Pretty Things Jack D'or on tap... you must try it. It's a local american saison, big american hop character and lots of saisony spice. It is my favorite beer.
As for local brews to try
Fisherman's Pumpkin Stout - Nice roast character, very subtle spice, one of the only pumpkin beers I like.
Harpoon IPA - Local staple, not as hoppy as the west coasters (~50 IBU), but great floral hop character, closer to british style.
Pretty Things Jack D'or - Only sold in 22ozs or on draft... awesome
Allagash Tripel - not sold in 6 packs I don't think. Great tripel. If you like wit's, there's is great.