My experience in Sicily is only with my own car. Driving in Sicily is not any different than anywhere else in Italy, but I know that some Americans, for whatever reason, don't feel comfortable driving in Italy or anywhere outside the US, and so I don't know if you would be one of them. Automatic cars are available in Italy, but they tend to be limited to larger models, and rarely come in the economy size models. Garmin works, TomTom works, paper maps and road sign work too (that's what I used in Sicily, since I didn't take my GPS with me).
Nevertheless public transportation is an option too. Train service and buses are also available, as well as boats to the smaller islands (e.g. Aeolian or Lipari islands).
Taormina to Mt Etna:
The best is an organized Jeep excursion from either Taormina or from the Rifugio Sapienza (which can be reached by car). There are several agencies in Taormina where you can inquire. Obviously there are no trains going up there and you will need an off the road 4x4 vehicle to get to some places.
Taormina to Syracuse:
Can do by bus or by train. Both work, but I think the train might be a bit faster (less than 2 hours).
The Taormina train station is down the hill in Giardini Naxos.
www.trenitalia.com
For bus service check below:
http://www.interbus.it/Home.aspx
As suggested above you could visit Siracusa from Taormina on a day trip, if you want to save yourself a hotel change. But plan on 2.5-3 hours each way, and that will cut down severely on your time available for visiting Siracusa, which is the most beautiful city in Sicily, IMAO (in my arrogant opinion). It's up to you.
Palermo to Monreale, etc. Is bus or train better?
No trains can go uphill. Monreale is on a hill overlooking Palermo. Take city bus no. 389 from Piazza Indipendenza (near Palazzo dei Normanni).
Would you suggest a tour company for any of these?
Mt. Etna (see above). Everything else is easy on your own.
Any tips on the transit there?
No particular concerns, just try to buy for the bus/train ticket. I know it's Sicily, but at least visitors should pay, since locals don't always do.
One more thing. When on crowded buses, be alert for these and these, and not only in Sicily, but everywhere.