I visited all those places except the islands in about a one-week period by train - easy. Even went to Seegesta (sp?) west of Palermo by train though bus would be better.

Look for a bargain Sicilian train pass sold at local stations. Sometimes the extensive bus system can be better or quicker than trains.

You can't use Taormina as a based for any of those cities if you mean doing daytrips using public transportation. Even with a car, it'll be very very long days for any of those places.

You need to go from one place to the other, going the around the island in more or less a loop, staying at each place 2-3 nights.

When we first "did" Sicily we started in Taormina for three nights, then two nights in Siracusa, then one night in Agrigento, then three nights in Palermo. We had planned to do public transportation, but when we got to Siracusa, it seemed next to impossible to get to Agrigento without taking all day and getting there late at night.
So we ended up renting a car for just one day, picking up in Siracusa and dropping off in Agrigento to take the train on to Palermo. We were glad we did as we enjoyed that drive and making a few smaller village stops along the way. And we were in Agrigento in the very early afternoon, giving us lots of time there. The car was really cheap, and would have been even cheaper if we had taken a SMART car, but Lee didn't really want to drive one of those -- new to us at that time.

My route, all by train was:

night train Rome to Palermo

Palermo - 3 nights
short trip to Cefalu for a few hours - great seaside town that in retrospect i'd stay in instead of Palermo

day trip to Seegesta temple (again bus would be better but i took train and walked a few miles along a road)

train to Agrigento - 1 night - got there by early afternoon - saw temples and even went down to the seaside resort

train to Taormina - 3 nights

day trip to Mt Etna - did the CircumEtnea narrow-gauge train around the base - slices thru lava flows, etc - at times tracks are indeed cut

day trip to Syracuse

night train back to Rome

I have been to all those places but not on the same trip. I like to start & end a holiday with a few days just relaxing somnewhere beautiful. So what I'd do with 2 weeks, if I wanted to fit all those places in, even though it's not the most efficient geographically, would be...

Fly to Catania, bus/taxi to Taormina, stay 3 nights.

Hire car, drive to Siracusa & stay 2 nights. Spend whole day on archaeological park & catacombs & wander round Ortygia in the evenings (stay there if you can).

Drive to Agrigento seeing Piazza Armerina en route, stay 1 night, see the temples & the Hellenistic Roman Quarter the next morning.

Drive to Palermo, ditch car, stay at least 3 nights.

Get train to Milazzo, get hydrofoil to an Aeolian island - Lipari if you want to visit others, Panarea if you just want to relax. Stay remaining nights.

Hydrofoil back to Milazzo, train or bus to Catania or Palermo for flight out.

As PalenQ notes, you can visit all these places using public transportation. I chose to rent a car to go from Palermo through Agrigento to Siracusa because I wanted to stop at some other places that would have been hard to reach without a car. I agree with the others that selecting a single base would not make sense.

BTW, to get from Lipari from Taormina, take the train to Messina then a bus (not train) to Milazzo, where you can catch a ferry. The bus station is only a few blocks from the train station in Messina. In Milazzo, the bus stops right by the harbor, but the train station in some distance away.

I hope you enjoy Sicily as much as I did!

I prefer trains but if i were advising someone i'd probably say a car is better for Sicily as the train system is rather skeletal and often you can't get there from here.

And with a car you can hop on the freeway to nowhere that i saw from the train heading towards Agrigento - one of the famous pork projects that incredible has a freeway that just ends in the middle of nowhere.

Oh, Palenq, please don't tell us Europe has those things too. I thought we Americans had the market on those silly highways and bridges to nowhere pork barrel projects.