The weekend farmer's market, held in the plaza in front of the United Nations University on Aoyama-dōri, is Tokyo's best. Dozens of farmers come from surrounding prefectures bringing organic fruits and veggies that, for a generation of Tokyoites weaned on supermarkets, is a revelation. There are also vendors selling tea, pickles, preserves and other souvenir-worthy goods.
The market is as much a place to shop as a social scene. There are always several food trucks, along with picnic tables for eating. As of the time of writing, the best one to try is Komeshiruna.
Events are often tacked on to the main event, including regular coffee, wine and sake festivals. Check the website for info on these and other events.