I have been to Greece a few times, and have learned the language. Athens is beautiful. There is a restaurant called Margaro, on a random street, and you have to be careful not to miss it. They only have one kind of wine but the best prepared mullet. The Greeks have been eating this fish since ancient times.
Also try the loukoumades - I think the cafe I would recommend is closed, but you should be able to find a place for it since it is a Greek staple.