When I lived in the sq, I used to trek just a tiny bit West on Fullerton to the Jimenez and Tony's.
There are actually a BUNCH of little fruit markets and taquerias throughout, you'll be trying plenty if you dare. Las Asadas on Western will probably be your friend if you like steak tacos, tho. Great neighborhood. Close to everywhere, and there's an on ramp to the Kennedy 50 yards from wherever you are standing at any point, it seemed. Used to love that Logan Blvd exit, the California exit, the Belmont exit...Target, the Jewel up around Addison, and there used to be a Del Rey (fruitmarket) and a Dominick's right on Belmont and Kimball (ish) which was right down around the corner from Chicago Food (Joong Boo - but it'll always be Chicago Food to me, sorry.) For produce, tho, if you have a car, my suggestion would be to take the really quick trip down Fullerton to Tony's or Jimenez.
We cannot be friends if you do not know the difference between Mayo and Miracle Whip.