Hello Carmen. Can't help with info on festivals I'm afraid. Perhaps someone with more local knowledge can help on that one.
As for restaurants you can't really go far wrong. For a traditional taverna type meal my own choices would be:-
In Yialos - O Meraklis (in a street at the back of the harbour) and The Trawler (Trata) (in the square at the bottom of the Kali Strata)
In Chorio - Syllogos is good with some 'different' dishes and you must not miss the famous Georgios. (Both on the square in Chorio)
We also love Tholos (on the point past Harani and before Nos beach)
As I say, this is just a personal choice. There are loads of good places and you can go very much up market if you wish. If you pick up a copy of the Symi Visitor when you arrive they have quite a useful map in the centre pages which show these and many more.
Have a great holiday.
Sue