Well done on trying to learn greek. Fruit and veg is pretty much a staple of any diet in Cyprus and the best place to buy it is at your local Froutaria. In my opinion, the quality is usually better than in the big supermarkets as the produce comes from local fruit and vegetable growers, usually picked straight from the fields that morning.
Most of the staff at the Froutaria speak reasonable English, but it's always helpful to learn Greek - fruit and veg signs are usually written only in GREEK so take a hint...
I really like to practice my Greek when shopping and I'm always learning new words as since we arrived here back in 2005, the range of produce on offer has grown enormously.
Local farmers have diversified into growing exotic fruit like passion fruit or dragon fruit, not that I know the name for the latter of the top of my head!
Learn Greek - Fruit and Veg
Just like on my other learn Greek pages, remember to put the stress on the part of the word in italics
If your local froutaria (fruit market) stocks types of fruit and veg which are not on this list, don't be afraid to ask for the word in Greek so you can increase your vocabulary and learn even more Greek fruit and veg names.