Taormina

I don't know where the port is, but there is a (public) bus that goes up the hill and connects to the train station, as well as a gondola that runs from the beach below. If you can get to the beach below, that would be ideal. I wouldn't walk up the winding road, it probably won't be too pleasant with speeding vehicles and vespas. Once in Taormina, everything is easy, compact and walkable.

Look on the map at the end of this article. The gondola is called "funivia" in Italian. "Stazione" is where the train station is located.
http://www.planetware.com/tourist-attractions-/taormina-i-si-ta.htm

There are frequent buses from the Taormina-Giardini Naxos train station up to Taormina itself. Last year one simply paid the bus driver. Have a map with you so you can navigate the town after the bus drops you off. It doesn't go into the middle of the historic area, which is pedestrianized. Taormina is built on the side of a hill, so there are changes of level to be negotiated, but it is certainly walkable. You can very easily manage this trip on your own; just be very clear on where you pick up the bus to get back down to the train station and what the schedule is. The bus schedule may well be posted inside or outside the train station.

I'm not familiar with the port in Catania, but the train station is along the coast, so it shouldn't be too far by taxi. It might even be walkable, but the entrance to the station is on the land side of the tracks, so you'd have to plan ahead of time where you were going to cross the tracks.

A taxi from Catania to Taoromina will cost anywhere from €65-85 each way and take about 40 min. You should find plently of taxi's waiting at the port, just be sure to negotiate the price before getting in.

A cheaper option would be to take the bus from Catania to Taoromina. The trip takes about an hour and there are buses hourly, costing €5.10 one-way, or €8.50 round-trip. See http://www.interbus.it/ for schedules. You can take a short taxi ride from the port to the Terminal Bus on Via Archimede. The bus drops you at Via Luigi Pirandello in Taoromina, just 400 meters from the Porta Messina, the NE entrance into the old town. The old town is very walkable.

You can also take the train from Catania to Taoromina, but the station in Taoromina is about a half-hour walk into the old town. There are local buses connecting the station and the old town, or you could take a short taxi ride. From Catania's port, you can take a short taxi to the Catania Centrale train station, where there are trains hourly to the Taoromina-Giardina station. Tickets cost €4.30 and the trip takes 45-60 min., mostly by regional trains, so it's easy to buy tickets on the spot at the station. Be sure to validate your ticket in one of the boxes in the station or on the platform before boarding the train or risk getting a fine (easy to do, just watch the locals). You can check Italian train schedules and prices through Trenitalia, the Italian national rail website: www.trenitalia.com/tcom-en

Thank you everyone. Re the taxis, we are five travellers so is the price per person or will the taxi charge for the trip be at the estimated 65 to 80E no matter how many passengers. Do they have taxi vans. We can also split up. On return, can we prebook a taxi?

With that many people, I'd be inclined to try the taxi route, at least one way, because of the timed saved. Perhaps you'll be able to find a taxi large enough to accommodate you all since you won't have luggage with you.

One additional point about the bus station in Catania: At least as of summer 2015 the ticket offices (separate one for each company) were not at the bus station, but rather 1 - 2 blocks away. Figuring out which office is the correct one and then getting over to the station would take a bit of extra time. The bus station neighborhood isn't unsafe, but I'd call it a bit "scruffy".

If you get an early start and have extra time on your way back from Taormina, Catania's historic district is worth seeing.

The Catania bus station is outside to the left of the train station, about a block away. You (or one of you) can go straight across the train station piazza to buy tickets from one of the agencies (you'll have to check their schedules), then go to the waiting area for the buses. There are at least four different bus companies served by the bus station. Make sure you get on the right bus. Lanes are numbered and schedules/destinations are posted.

There are a few bars within 50 meters, in case you have some time to kill.