I am planning to stay 3 nights in Syracusa and use it as my base to explore Noto to Taoromina. From there, I was hoping to stay 2 nights in one of the islands, preferably Salina. Can anyone recommend ways to get from Syracusa to Salina? By car, or train is a possiblility but I don't know how far. Also, any hotel suggestions in Salina or a different island that would be better to stay to explore the rest of the islands? Any recommendation is appreciated.

I'm not sure Siracusa lends itself to exploring Taormina. It's about 1.5 hours by car, longer by bus (generally preferred to trains in Sicily) with a change at Catania airport.

Boats to the Aeolian islands leave from Milazzo. It's possible to get there by bus or train, but a car is simplest.
Lipari has the most frequent connections to the other Aeolian islands.

There are regular buses to the Milazzo ferry from Catania airport. You could drop off the car at the airport and then take the bus to the ferry; or drop off the car in Milazzo, I suppose. It would be an easy drive along the coastal highway.

I think the Ferry just goes Milazzo/Lipari. You would have to switch to the local island-circuit ferry to get to Salina.

In Salina, just to the left of the ferry dock (the first restaurant after the little walkway), there is a granita and ``loaded bread'' place. They serve the best granita in the world. Seriously. The Italians I was with were all sitting around trying to figure out why it was so much better than everywhere else; the students I was with ordered 3-4 servings apiece. The loaded bread is like a cold pizza with salad-like toppings. Also amazing.

If you want to stay in Lipari, I can recommend the Grand Hotel Arciduca (grand, but no water views), or the Hotel Giardino sur le Mer (spectacular water views). They are both about a 5-10 minute walk from town center.

Lipari has more going-on than Salina, but both are tiny.
Either would be a good place to use as a base.

To get to Salina from Siracusa, you will need to get to the port of Milazzo. From Milazzo you can catch the Hydrofoil to Salina. The only public transport to Milazzo from Siracusa is by train, which takes around 5 hours. You could rent a car, but don't really need it in Salina, since you can rent a motor scooter, which is more fun anyway.

There are some great places on the port of Salina:

www.hotelprincipedisalina.it
www.mammasantina.it
www.lasalinahotel.com

But my favorite place is the quiet place on the Pollara side of the Island...where they filmed "Il Postino", called Locanda del Postino: www.lalocandadelpostino.it

Make sure and get a granita at da Alfredo in Lingua, do a wine tasting at Capo Faro or Hauner for some Malvasia.

Maybe hit Taormina on your way out from Siracusa, then head to the Aeolian Islands from Taormina.

Have fun!

Milazzo is the main port for the Eolie. There are two main operators, Siremar (www.siremar.it) and Ustica Lines (www.usticalines.it) with about a dozen services daily to Salina.
You can get a train to Milazzo but the railway station is not in the town, so you'd need to take a bus or taxi from the station. It would probably be better to take the train to Messina, then the bus to Milazzo. The bus from Catania airport to Milazzo runs only once a day.
www.giuntabus.com/orari.htm

Thank you for all your helpful ideas. I will take a car to Milazzo. The simpler the better. Thanks for hotel recommendations. I will check all of them out.

Whoops. I just realized that that world's best gelato place is near the boat docks in Rinella, on the south side of Salina, not the main port town. But it is worth the trip.