Messina port

There probably are taxi at the port or certainly in front of the Messina train station which is 5 min walk away.

However a cheaper way to do it is to take the train to Taormina-Giardini from the Messina train station, then from there go to the city center of Taormina (on top of the hill) by bus or taxi (3 min ride). The train gets you there in less than one hour (the taxi about 45-50 min). Messina is about 60 km from Taormina, therefore the taxi is likely over 100€ one way.

See trains timetable:
www.trenitalia.com

From: MESSINA
To: TAORMINA-GIARDINI

Patty,

If you take the train, be sure you can get a return to port that will not make you late for your departure.

We looked into taking a train there as well. I don't recall that there were a lot of trains back to the port that would work for our schedule.
So, instead we decided on another tour option that will drive us in a van from the port and let us explore the city on our own. The van will then take us back to port and guarantees that they will get us back in time.
The particular tour we chose only allows 2 hours in Taormina, but that will work for us based on what we wanted to do there.

I was not sure if I wanted to do a 45 minute each way road trip and only be in the city for 2 hours, but decided it was worth it for the views from there.
Plus, the scenery on the ride sounds like that is also worthwhile.