getting from Rome to Palermo

Fly from Rome to Sicily (either Palermo or Catania--into one and out of the other may work best for you). Check skyscanner.com to see what your options are. I took the night train in 2015 and it was miserable because of (I assume) rough rails. I got not one wink of sleep.

There is both train service and bus service on the island. There is very little, if any, intercity bus service on Sundays and holidays, so check the calendar carefully as you lay out your provisional plans.

Trains will generally be fastest along the coast. There's also a line that runs through the interior between Catania and Palermo, by way of Enna. I prefer to use the Deutsche Bahn website to check European rail schedules, but you'll need to use Trenitalia to find fares. Trenitalia insists that you use the Italian station names (so "Roma Termini" rather than "Rome").

To check bus schedules in advance it is often easiest just to Google "bus Palermo to Trapani", etc. If that doesn't work, you can go to Rome2Rio.com and drill down through the results until you find a link to the appropriate bus company. Rome2Rio is also the place to start if you're not sure what mode of transportation to use on a particular leg. Just don't trust its fares, travel times, or frequencies.

Five years ago, after a week in Rome, we caught an EasyJet flight to the Palermo airport. Cheap, fast, easy.

The airport is actually several miles west of the city of Palermo itself, and we never actually visited Palermo. We got a Europcar rental car at the airport and drove around Sicily for the next week. It didn't appear that public transportation was going to be a viable method for us to see Erice, Agrigento, Siracusa, Taormina, or the interior of Sicily in the time we had. The highways were toll-free, and had very little traffic in December 2012 - very easy to get around. Streets in towns were often narrow, however, and parking spots weren't always easy to find. Maybe you're determined to use public transportation, however.

As Rick Steves didn't have a lot of Sicily information at the time, our primary resource for travel information was the Lonely Planet Sicily guidebook.

I am going to Sicily in February. I just booked flights from Rome FCO to Palermo on Ryanair. We leave out of Catania to Pisa on Ryanair. These flights are direct, a little over one hour and inexpensive.

thanks very much for all the info!

How many nights do you have?

Flying is the best way.
To see who flies from Rome to Palermo, go to the Wikipedia article on the Palermo Airport (also known as Falcone-Borsellino Airport, from the name of two famous prosecutors killed by the mafia in the 1990's).
While in Sicily, you can travel by bus or train, depending on the destination.