Hi Garry welcome to the chat page.
You will love Taxiarchis. I hae stayed there many times and coincidentally can see the place from where I am sat right now topping up my suntan
All the tavernas in Chorio are worth a vist. Zoe's at the top of the hill from Taxiarchis is a good place to start - ask Frank to show you the way. Then there is Syllogos and the famous Georgio's to try. For something a little different the Windmill restaurant serves interesting and delicious mezes. For a snack or light lunch the Olive Tree cafe is a must. Finally the often overlooked Manteio bar close to Zoes serves good snacks for a very reasonable pricealnog with a fine selection of unusual beers.
All these places are within ten minutes walk of Taxiarchis. If you venture down to the harbout there is a far bigger choice but I will let others recommend those, I am posting this from a Kinde and it is slow going!
Symi is a wonderfully picturesque place so there will be no lack of photo opportunities.
I hope you enjoy your holiday.