Taormina or Termini Imerese, Sicily?

Hello karencooklac. Rick Steves does not read the questions here. This Travel Forum is a public board. Any person can reply to questions here. I have no knowledge of Termini. I recommend going to Taormina . It is a good place to be at for 3 nights. Hotel Villa Belvedere is very good. It is at the top of a cliff, very high above the Mediterranean Sea. That hotel's swimming pool is at its patio (terrace). A buffet lunch is served to the guests at that patio (terrace). Walk to the cable gondola (lift) for a ride down to the beach (Mediterranean Sea). I do not know about swimming in the Sea at that beach. Taormina has many trees and green plants. The airport is to the south of Taormina. An other hotel, similar to Belvedere, at a near by location and similar setting (the hotel has a patio or terrace at the top of a high cliff above the Sea) does not have a swimming pool, and the cost for being a guest at that hotel is much lower. Information and very many photographs of Hotel Villa Belvedere is at www.villabelvedere.it
(Click on little British flag for English language). Very many photographs of that hotel at Photo Gallery in that internet website. Those photographs can be enlarged for your computer's screen.
www.tripadvisor.com/Hotel
www.booking.com/Italy
www.hotels.com/Taormina

You can do both, but considering your limited time, you should rent a car in this case.
To do so you should fly with an early flight from Rome to Palermo, which is closer to Termini Imerese than Catania (the other main airport in Sicily).
Once you arrive rent a car and drive East toward Palermo and beyond Palermo. Stop in Termini Imerese for a while to visit your ancestors' town. There isn't a whole lot to see in Termini, so a few hours will be enough for you to say you've been there.
After you are done, proceed East on the A19 freeway to Catania or the A20 to Cefalu/Messina to go to Taormina. I prefer the A20 because it's more scenic and it avoids the Catania traffic. Along the way, you could also stop in Cefalu, which is very pretty and has a nice beach.
Once in Taormina, stay there. If you want a beach, you might opt for Giardini Naxos, at the bottom of the hill where Taormina sits.
To return to Rome, take a flight from Catania airport, which is closer to Taormina.

Hotel stars are not so useful in Italy. They indicate things like whether a place has an onsite restaurant, but not whether or not it's comfortable, plush, a dump, etc.

Here's my review of my Taormina accommodation, the Casa Cuseni, which was definitely NOT a dump. See if it's what you're looking for - I'd stay there again in a heartbeat: https://community.ricksteves.com/travel-forum/italy-reviews/taormina-accommodation-casa-cuseni

Taormina is magic, and a marvelous place to unwind; the views and gardens at Casa Cuseni really amplified this, and made my stay better.

I'll second Harold's recommendation of Taormina. We spent four days there in October 2010, and even with rain a lot of the time, the vistas were magnificent & the food--which I understand is a personal preference--was more to my tastes than anything else in Europe to date. Seafood & vegetable based, with a lot of 'yin-and-yang', sweet/sour, almost German-influenced at times. Interesting cuisine.

We stayed at Hotel Villa Schuler, which was wonderful, and had memorable meals at Rosso Peperoncino and Osteria Nera D'Avola.