Ok, grocery shopping isn't necessarily a hip thing to do. It is however, a required part of life due to our annoying need to consume food to survive. So, if you have to do it, you might as well get the most out of it and make it as painless and as cheap as possible.
In the New Haven area, to turn shopping from an expensive boring chore to something else, you basically need to know about 2 stores, Trader Joe's and M & M Farms Produce of Milford.
Trader Joes is a national chain that seems to approach the idea of the basic grocery store in a very different way. First of all, their prices are very reasonable, much better than larger chains such as Stop & Shop or Shaw's. More interesting to me though is their great selection of organic and vegetarian food options, and their huge variety of packaged foods that you just have to heat and serve. There is a strong emphasis on non-western culinary traditions, so you can easily get things like bags of Edamame or Papadam chips. If you aren't eating organic meets and dairy, you should be, and Trader Joe's can help you get on the right path without having to spend a ton of money (you'll save money versus normal grocery stores, and end up with better food). Finally, Trader Joe's is great due to its laid back and friendly employees, all dressed in colorful Hawaian shirts. Trader Joe's is located at 560 Boston Post Road in Orange.
If you live in the New Haven area, you should never buy fruits and vegetables from anywhere other than M & M farms, located on 804 Boston Post road. They have the best produce at prices that are less than half of what you would pay at a chain grocery store. You can also get fish, baked goods, or deli meats at great low prices. Finally, they have a great asian specialty store where you can get a wide variety of south and east asian spices and foods (everything from japanese candy to indian fast food).
Go forth, eat and be merry! For less!