We have several Farmers Markets around Austin. I just went to the one in east Austin yesterday. It was a lot of fun, as they had live music and several vendors selling already prepared food and drinks in addition to the fresh produce for sale. If you want fresh eggs and meat you better go early as these items often sell out early.
Here are links to the ones I go to the most often. Have fun and eat local!
http://www.hopefarmersmarket.org (East /downtown Austin)
http://www.bartoncreekfarmersmarket.org (Southwest in Barton Creek MAll parking lot. This spot is also great for getting a view of downtown Austin)
http://www.jbgorganic.com (open M-Thurs 8:30-4:30 and Friday 8-12) Johnson's Backyard Garden)
http://TexasFarmersMarket.org (Cedar Park)
http://sfcfarmersmarket.org/#myGallery1-picture(4 locations around town)