I would love to explore the town and surrounding areas. Would prefer to stay in a 2br house or condo (or suggestions) . Can you rent a car there or is there public transportation?
Perhaps old school, but I am going to vote for a stay in a hotel.
Staying in the hotel, unlike the apartment, let you always have a bearing on the staff at the front desk that can give you all the tips for organizing the visit.
Taormina is a very small city and it may seem easy to be able to see it all by yourself, but if you are interested in a real travel experience, to really know the Sicilians you have to talk with them, then pluck the opportunity to get as much guidance as possible.
The hotel staff can help you to rent a car to reach Mount Etna and the Alcantara River Park ( http://www.terralcantara.it/en ), for example, or suggest you what presents to buy and get a discount for you (Sicilians are great dealers).
You don't write your budget, so here are different suggestions for different hotel's categories:
5 stars L - http://www.grandhotelsanpietro.net/
4 stars - http://www.hotelvilladiodoro.com/
4 stars - http://www.hotelcaparena.com/
3 stars - http://www.hotel-isabella.it/ about 8 years ago