If there's one unfortunate thing that I've learned, it's that it's entirely possible for someone to live here in a little English-speaking, Hemingway café-going, Bon Marché shopping bubble. The sheer diversity of lives within this microcosm is incredible, and just dipping into the shallow pools of those of the wealthiest lifestyle can only leave you shivering with cold. Those who follow such a lifestyle don't know this diversity of this place, and to their detriment, they never will.
1) Belleville Market
Le Marché de Belleville
Tuesdays and Fridays, 7:30-2pm
63 Boulevard de Belleville
75011 Paris
Métro: Couronnes
2) Canal Saint-Martin
Canal Saint-Martin
Métro: Juarès
3) Le Parc des Buttes Chaumont
I have yet to go, but there is apparently a really neat bar here called Rosa Bonheur. So if you're in town, check it out!
Parc des Buttes Chaumont
Métro: Laumière, Buttes Chaumont, or Botzaris
or, take a stroll up the Canal Saint-Martin and a make gradual right turn to get there.
So my advice to any visitors here (for longer than a week) is the following: Take a Tuesday or Friday, get up and head out to the market, buy some nice fruit (you won't get much cheese or prepared meats here, but just stop by one of those nearby, always ubiquitous boulangeries to buy some sandwiches for lunch!) Gather your picnic materials, walk along the Canal Saint-Martin up towards the Parc des Buttes Chaumont, and make a day of it there... lay out your goods on the grass and be ready for some good conversation, meditation, or reading time!
Enjoy!