If you’re into organic food, Trader Joe’s is a natural choice. There are two relatively near, one near the Fresh Pond, the other one on Memorial Drive. Even if you don’t care about organic food, they have an interesting selection of wine and beer – more affordable than in the regular liquor stores, too!
Buying groceries online in Cambridge is quite easy, and you do it on Peapod or on Instacart, as well as on Amazon Fresh (it has an annual membership fee, but delivers for free, there is no tipping hastle, and the prices are ok).
If you want healthy, organic food, Whole Foods is pricy but way more green, organic, healthy and environmentally-aware than most other corporate supermarkets. If on top of organic you want co-ops, there is a Farmers Market every Wednesday in Oxford St (in the core of Harvard). Harvest Co-op Markets, in Central (Cambridge) is a co-op with a wide diversity of products (and you can use Instacart to buy from it online). And there is a community-supported agriculture collective (Farmers To You) which provides locally grown organic food.
When shopping for food products, keep in mind that most American products are sweeter, at least as compared to Europe (to the extent that many Europeans can’t stand eating an American fruit yogurt). Also, American companies use corn syrup instead of regular sugar – it is cheaper, but makes you gain weight much quicker. Remember also that while Americans are relatively aware of fat contents (and often would advertise food as “fat free”), they still fall silent on sugar contents. When Dariusz wasn’t watching his weight in the US, over a year he gained 9 kg, so beware!
Btw, two-cents: It is really easy to make your own Kefir or Yogurt (about 8 times cheaper than to buy, too).