The Central Markets on Athinas St. are worth spending some time. You'll be right in there with locals and taverna owners who shop for their daily food and what to serve in the restaurants.
The Meat/Seafood section is on one side of the street and if you are squeamish about meat then you may want to avoid that section but it's still worth going to and seeing something that you'll never see in the US or most other Western countries. I won't be too descriptive of what you'll see but it won't be going to your local supermarket and being nicely wrapped meats in styrofoam containers!
The seafood section is really special with lots of seafood you'll never know existed.
Just a hint about the seafood/meat area . . . don't wear sandals.
Across the street is the produce section which may be more appealing to most people. A wonderful selection of veggies, fruits, dairy, with a number of small shops selling prepared foods, baked goods and lots more.
In addition there are numerous street vendors in front of the meat/seafood area selling dried fruits, nuts, herbs, spices and other goodies.
In and around the Central Markets are numerous small, locally owned specialty shops that are amazing.
It's not really a tourist destination but that may be the best thing about it . . . you'll be dealing with locals rather than pushy and shoving tourists!