Palermo to taromina

Is it possible for you to fly into Catania instead? It's an easy bus ride from the Catania Airport. It seems like you're zigzagging a long way if coming from Palermo. Do you have plans on seeing any other places in between or on the way? Either way, I would take the bus (with the caveat that Catania is a more logical entry point to Taormina).

Train is also possible; you'd normally travel along the northern coast to Messina and then switching to a southbound train. In theory you could take a train through the interior to Catania and transfer there, but I believe that route is served less frequently and would take longer. The advantage of the train is that you can pack a picnic lunch and be ready to start sightseeing shortly after you arrive in Taormina.

The nearest train station to Taormina (which clings to the side of a hill) is down at sea level in Giardini Naxos, so you end up having to take a local bus up the hill to Taormina anyway. I'd base the train-vs.-bus decision on total travel time and convenience of the departure time. It's nice to be able to move around a bit on the train, but the buses are generally comfortable.

Train schedules are here: https://www.trenitalia.com/en.html

If you decide to take the bus, be aware that as of 2015 (I don't know whether it has changed) there wasn't a bus station in Catania where all the companies had ticket counters. Each company had its own small office within a couple of blocks of the open-air parking lot from which buses departed. This wasn't a problem as long as you were aware you'd need a bit of time to get to the other company's office (unless you had purchased your ticket ahead of time). I believe you'd be changing bus companies, not just buses, in Catania.

So helpful. Thank you
We are starting at Palermo then to Taormina, over to Puglia (Lecce) then to Amalfi. Nothing in stone. We are NOT seasoned travelers but trying to wing it although we are 75.

This is totally unsolicited advice but you mentioned you are not seasoned travelers..so please feel free to ignore. I think you may be trying to do way too much jumping around very disparate places, which could get quite tiring. You can have a lovely trip just in Sicily alone without jumping from the mainland to Sicily and then trying to cover multiple sides of Sicily (and then two coasts of mainland Italy too). There are a LOT of "in between" and interesting places to see, so you can potentially have a vacation that does not have so much transportation time getting from one place to another (with sites/ attractions that are more geographically compact).

How much total time do you have on this trip? Do you have a copy of the RS Siciliy and Italy books?

From Sicily to Lecce is a very long trip. I connected the two in 2015 and opted for a bus instead of a train because the bus was supposed to be faster. Unfortunately, the air conditioning on the bus was found to be broken before it set out on the trip (this was in the summer). They spent several hours trying to either get it fixed for obtain a replacement bus. Having no luck, they took the best action they could, which was to send us the bus with no a/c, at which point our departure was several hours late. It was not a fun day. (Of course, the a/c occasionally breaks on trains, too.) It would have been even worse if I hadn't positioned myself in Messina the night before departure in order to shorten the trip a bit.

I'd highly recommend planning a stopover somewhere along the way between Sicily and Puglia.

Alternatively, you could consider flying. RyanAir seems to have one flight a day (not always at the same time) between Catania and Bari and a separate flight each day from Palermo to Bari. Both are non-stop. I can't find any flights to Brindisi. Choosing to take a RyanAir flight means you must read its rules very, very carefully lest you end up being charged horrendous unnecessary fees. Proceed with caution.

Sicily needs at least two weeks. You are short changing both Sicily and Puglia. Give this trip more time or drop a location.
They are not expensive destinations so add several weeks to this trip.