Have been in Boston for the last 13 years and I love to eat and drink so....
For craft beer bars, honestly there's a lot of good options now. Most higher end restaurants will have a decently curated small selection. The best beer + food (seafood) combo as
@HuskyHawk mentioned is Row 34 in the Seaport, especially for sour beer (and convenient to picking up Trillium beer to go) and oysters. There are good clusters elsewhere such as in the Kendall Square area of Cambridge. You have Lord Hobo (bar not brewery), Cambridge Brewing Company, and Meadhall. Plus Smoke Shop for pretty good BBQ. If you find yourself more in Brookline or Allston area, you've got The Publick House, Lulu's (might have best selection in Boston), and Deep Ellum (RIP Sunset). For downtown, I'd lean towards The Gingerman, Stoddards, or the Tip Tap Room.
For food, if it's a special occasion or you're willing to splurge and you're adventurous eaters, my top 2 food "experiences" in Boston are The Table (by Top Chef contestant Carl Dooley) and Tasting Counter. These are both intimate, high concept, multi-course paired beverage food adventures being prepared in front of you.
For best food and still expensive but maybe one step down in experience from those two, I love Craigie on Main (farm to table meets classic french), Sarma (Mediterranean small plates), Oleana (Middle Eastern small plates, good for vegetarians), Alden & Harlow (farm to table small plates), and Bar Mezzana (Italian and crudo).
For something a bit more casual, Brewer's Fork has a nice combo of Pizza + Beer and is pretty close to the USS Constitution and end of the Freedom Trail. You could also do the aforementioned Lulu's, get some ramen at Yume Wo Katare (be aware of lines), or if you're near Kenmore try Sweet Cheeks BBQ, Eventide for a lobster roll, or Tasty Burger for a divey burger + smothered fries.
If you want to visit a brewery, Night Shift in Everett is the best vibe + beer combo, although a little bit out of the way with no car. The good news is that there is another brewery (Bone Up) and a distillery (Short Path) within walking distance once you get there. Someone mentioned the Trillium Beer Garden on the Greenway and that's a great choice on a nice day, especially if you like IPAs. Lamplighter is a pretty cool spot and their beers have gotten quite good.
If cocktails are more your thing, I'd recommend Yvonne’s downtown, Blossom Bar in Brookline, Waypoint (sister to Alden & Harlow, good food here, too), Back Bar in Somerville for a more hipster experience, or Drink in the Seaport.