Besides the previously mentioned supermarkets, small fruit and vegetable shops called verdulerias are found on almost every block here. Each has a selection of maybe 10-20 products, and they pick the items for you, rather than the self-serve style at the supermarket. (I haven't verified this yet, but I assume this allows them to be a little cheaper than their bigger competition.)
One of the standard vegetables sold at every verduleria is a green vegetable called a zapallito. We bought it as an experiment and decided that it's basically a round zucchini.
The other day, I combined a zapallito with a red pepper (which are enormous here), onion, chicken, and rice, for a very hearty stir-fry.