There is a well-located market next to a lovely green park in the VII. district. You can find fruits, vegetables, pastries and cheese, meat etc. under the pavilions next to the park at Hunyadi tér. Watch out, there is also a market hall at Hunyadi tér, but what I would recommend is the outdoor one, right at the park.
On Saturdays, there is a whole variety of local producers selling their goods all morning. You can also find some local fruits and vegetables on weekdays with less variety, but the same quality. I love to walk around and buy fresh produce directly from the people growing it. After popping up there a few times, they often recognise you and have a few kind words while choosing the prettiest products for you. This is when a city starts to feel like home to me.