A must see is Warowot Market on the east side of the city near the Ping River. It has everything from household goods, to meats and vegetables, all the way to monk kits (these are starter kits you purchase to give to a monk that contain many different sundries such as rice, vegetarian snacks, deodorant, toothbrushes, etc.).
The Night Bazaar and Anusarn Market (open every evening from 6 - till) have all the basic souvenirs you could want. The Saturday and Sunday "walking street" markets have more of the same but are located in different areas of the Old City. Some markets specialize in handmade crafts or other local art work. Essentially, there is a market always open to peruse any day of the week until late in the evening.
One of my favorite markets is the one that comes right to my front door. A couple of times a week, a truck pulls up in front of our condo and opens its doors. They sell everything from fruits and vegetables to farm fresh eggs and toothbrushes. It's a little family-run operation, but I always like to buy from them. They are very friendly and I always get a great price.