If you know what you are doing going to a big shared room in a hostel you will be fine. There is a nice kitchen with place to cook and big fridge to leave your stuff. The place is well organized, there are even stickers you can use to mark your food/drinks in the fridge (there is a rule - thing is not marked is for everyone). The big terrace with many tables is very nice, plenty of room and good view of the city.
After visiting Taormina and spending 1 night there I can recommend to go there in the morning to avoid the crowds of rich Russians in the evening and spending your evening/night in another place nearby. It will be cheaper and there is not much to do in Taormina in the evening/night.