I love shopping - for food that is, not clothes (for the latter I have to be dragged kicking and screaming). I've come to rely on so many large and small shops in Boston area, that I can't even imagine cooking without them. Note that the hours of operations often changes, so make sure to call the store for the latest information.

All-purpose:

Whole Foods – organic supermarket chain that is great for one stop shopping. It's more expensive than a regular supermarket, but the quality and selection is better. Go to http://www.wholefoods.com for hours and locations near you.

Fish:

New Deal Fish Market – the best fish selection and quality in Boston, particularly for whole fish. They carry sushi grade salmon, tuna, branzino, hamachi, fluke, etc. Carl Fantasia (the owner) is a walking encyclopedia of fish and a fabulous cook always willing to share his knowledge and recipes. The fact that this little gem survived the Whole Food invasion and consumers' drive to have everything fast and cheap is a joy and a miracle. Just be patient. Every customer has her turn, and as the Miracle Max said in Princess Bride, "You can't rush a miracle, sonny."
622 Cambridge St., Cambridge, MA, 617-876-8227
Hours: Mon 3pm-7pm, Tue-Fri 10am-7pm, Sat 9:30-6:30, Closed Sunday

Wulf's Fish Market – The last great fish market on the Boston side of the river. Excellent quality and selection of fish. Alan Wulf (the owner) is extremely knowledgable, friendly, and will be happy to answer all your questions, or clean a whole fish for you.
407 Harvard Street, Brookline, MA, 617-277-2506

Captain Marden's – this is where the restaurants get their fish, so the quality is consistently high. The retail store only has fillets, but they can get you whole fish and all kinds of interesting species if you call a few days in advance.
279 Linden St, Wellesley, MA 02482, (781) 235-0860
Hours: Mon - Fri 8am - 6pm, Sat 8am-5:30pm, Sun 10am - 5pm

Frankie's Catch of the Day – wonderful little fishmarket. They only carry fillets and steaks, but the quality is top notch. If you want whole fish, call Frankie and place an order.
19 Leonard St, Belmont, MA 02478, (617) 484-6460

Whole Foods – most stores have a good selection of fish (including whole fish). The quality is consistently great in all the stores that I tried. My only pet peeve with Whole Foods is that they sell many skin-on fillets unscaled and do a poor job cleaning whole fish when I ask them to scale, gut, and take out the gills.

Meat:

Fresh Pond Market – This ain't Lobel's of New York, but their steaks are excellent and prices are reasonable. If you are lucky, you'll find prime, dry-aged NY strip for the best price in Boston. They also carry more unusual cuts like a lamb top round roast, and if you don't see something just ask. Chances are they have it, but it might be frozen.
360 Huron Ave, Cambridge, MA, (617) 876-3916
Closed on Sunday

Bread:

Iggy's – great breads and croissants. You can get some of their baked goods at Whole Foods and Shaw's, but the real bakery in Cambridge offers a wider and fresher selection.
130 Fawcett St, Cambridge, MA 02138, (617) 491-7600
Hours: Mon-Fri 7am-6pm, Sat 7am-5pm, Sun 7am-3pm

Clear Flour – great breads and even better croissants than Iggy's (if such a thing is possible). We rarely make it to Brookline, but often buy their baquette and focaccia at Whole Foods.
178 Thorndike St, Brookline, MA 02446, (617) 739-0060
Hours: Mon-Fri 8am-8pm, Sat-Sun 9am-7pm

Sel de la Terre – great breads, particularly baguette and fig anise bread. We rarely make it to Boston, but often buy their bread at Formaggio's Kitchen in Cambridge.
255 State St., Boston, MA 02109, (617) 720-1300
Hours: Mon-Fri: 7:30am - 9pm, Sat - Sun: 9am - 9pm

Produce:

Russo's – they have 7 types of eggplant. Need I say more? This is where restaurants get their fruits, vegetables, and herbs. Besides excellent, inexpensive produce, the retail location also offers ethnic products, prepared foods, flowers, and has a great deli, cheese, and dairy departments. The meat selection is very limited, and they don't carry fish (besides smoked fish sold in the deli).
560 Pleasant St., Watertown, MA 02472, (617) 923-1500
Hours: Mon-Sat 8am-7pm, Sun 8am-6pm

Farmer's markets – you can get the schedule or Boston area farmer's markets at http://www.ams.usda.gov/farmersmarkets/States/Massachusetts.htm
Most of them operate from June to October. They are not necessarily cheaper than regular supermarkets, but this is your chance to get local produce that is so much more delicious than the stuff we eat year round.

Pantry items:

Trader Joe's – great for olive oil, balsamic vinegar, and cheap cooking wine (Charles Shaw's and La Boca). Go to http://www.traderjoes.com for hours and locations near you.

Gourmet and Specialty Stores:

Formaggio's – very pricy, but fun store that offers excellent selection of cheese, deli, breads, and baked goods (their chocolate dipped biscotti are superb), and some produce and flowers. Great place to learn about cheese -- on weekends, they put out 6-8 cheeses for people to taste. They also sell Moroccan preserved lemons, which I find very convenient when I forget to make my own (they don't put them out, so you have to ask them).
They have 2 locations: one in Cambridge Huron Village and another in Boston's South End. For more info, go to http://www.formaggiokitchen.com.

Savenor's – They carry it all -- rabbit, foie gras, goat, interesting cuts of beef. At first glass, prices look high, but there are some real bargains to be found here, like reasonably priced hanger steak.
They have 2 locations: one in Cambridge and one in Boston. For more info, go to http://www.savenorsmarket.com/.