The best place to shop for food and other daily stuff is Vasiliadis supermarket. It's a large two-storey shop located to the right on the way from downtown Kos to Lambi. As far as I noticed locals do their shopping at Vasiliadis' as it's the cheapest and best supplied shop.
Anything you need can be bought on the spot. Greece is a relatively cheap country. I noticed that tourists from the UK found Greek tobacco products very attractive. A friend of mine who is a doctor was very interested in some medicines which are much cheaper there than in Poland.
This is a great little shop which i found is a fantastic place to buy your gifts to take back home with you beautifully presented with the tourist in mind
What to buy: This shop has everything you need under 1 roof to take back home as gifts. They sell lovely smelling soaps, shampoos, cosmetics, and real greek sponges
What to pay: The skys the limit