Most people know Szimpla Kert as the grandfather of all ruin bars in Budapest, but it's also home to an very popular and very fun farmer's market on Sunday mornings from 9:00 until 2:00. As I mentioned, it's very popular so get there early if you want to beat the crowds. We went at about 11:30 and it was packed!
Once you arrive, you'll find a couple dozen vendors selling fresh produce, homemade cheeses, artisan cured meats, honey, all manner of dips and spreads, pickled vegetables and olives, pastries, and incredible breads. Make sure to go hungry as most of the vendors also sell sampling platters (I can heartily recommend the sausage plate) and you can go ahead and have a fresh breakfast or light lunch while you're there. Szimpla's bar is open during market hours so you can also grab a coffee or a beer to accompany whatever tasty treat you've picked up.
Another thing I liked about the Szimpla Sunday Farmers' Market was the mix of people. Yes, there were a lot of tourists, but you also saw a lot of young alternative types there for breakfast as well as a lot of elderly residents from the neighborhood sipping coffees and nibbling fresh-baked pastries. It's a wonderful introduction not only to Hungarian cuisine but also to local neighborhood life.
My suggestions is that you head to the farmers' market right when it opens at 9:00 for breakfast and then head out to explore the nearby synagogues that open at 10:00 Enjoy!