For information about places to eat I recommend reading the various trip reports in our archive section:
http://symiforum.com/viewforum.php?f=19In particular, read the reports by "D and E". There is nothing they don't know about tavernas on Symi.
There are only two tavernas in Chorio now, sadly. Georgio and Maria's is just five minutes walk from your accommodation, Zoe's taverna a few minutes stroll further up the village "main street". Both are traditional Greek tavernas serving good food and worth a visit.
Around the harbour the traditional tavernas I like (and it's just MY opinion) are the Trawler at the bottom of the Kalistrata, Meraklis in the back streets at the head of the harbour, Spitiko and Harris tavernas side-by-side along the south side of the harbour and on the headland right round to the north past the boatyard in Harani is Tholos taverna. All good places serving traditional Greek dishes.