Hey! I'll be exploring Oaxaca & its surroundings with some friends next week for 3 days. We are easy going & friendly folks who love to talk with locals & eat authentic food at authentic places! Not an extended budget but happy to pay the price for an amazing experience :) Thanks!
You might want to pay a visit to the small village San Bartolo , Coyotepec which is situated around 7 miles south of Oaxaca.
this is the village where they produce the famous Barro Negro pottery and the village has several workshops and stores where you can buy the real stuff right from the producers.
A bit closer is the town of Santa María del Tule, Oaxaca which has an amazing cypress in the town square.
Teotitlan del Valle, Oaxaca, Mexico, is nearby as well and known for weaving.
My best recommendation is to try to experience any of these small towns on their respective market days. It is quite the experience with amazing food and sites.
Mentioned in this answer: