Restaurants There are plenty of restaurants in Athens. Some are good for their traditional meals some others for their nice view. In the latter category you will find restarants up to the lycabetus hill (quite expensive :standard course price is at least 20 euros per person). In the former category you can visit places in the plaka region exactly under the Acropolis hill. But you shouldn't except something unique as all of those places are used to survive on tourists and so their meals are meeting the quality standard of the souvenirs sold in the same region while their prices are not that cheap with a standard course costing up to 14 euros per person. I can recommend some cheap restaurants like: BAYRAKTARIS and SAVVAS for Greek souvlaki/kebab in Monastiraki (near Monastiraki Metro Station), KANELLA restaurant in Kostandinupoleos and Evmolpidon, Gazi (near centre of Athens), cheap and nice Greek food and plenty of small tavernas in Psirry area.
Supermarkets Major supermarkets chains can be found anywhere in Athens but not in the very center into which you can only find smaller mini markets. The quality of the goods is fine and the most known european brands can be found. The chain of AB Vasilopoulos supermarkets offers also cooked foods in high quality and low prices. Also there is a CARREFOUR in Abelokipous (centre of Athens) near President Hotel
Street Food The are quite a lot of choices for street food in Athens especially in the very center of it. Order a souvlaki (something like the turkish kebap) and don't accept to pay more than 1.5 euros for it-unless it's something special. There aren't so many street food sellers, but there are many small shops of chains like GRIGORIS, or EVEREST that sell nice sandwitches and it is a kind of street food.
Local Specialties There are many local specialities you can try to eat in Greece. Most of them are not really greek foods as their origins lie to istanbul and izmir and the immigrants that came from there in the 30's. So be prepared for an eastern influenced kitchen and order your musaka,kebab,kokoretsi,bamies,fasolakia,piazi,fresh sea food, horiatiki salata etc. Of course Greek foods are not so spicy like their oriental "brothers", but it worths to try them. There are very nice sweets and many varieties of sea food, if you like.