Travel from Sorrento to Palermo

I don't find overnight ferries or trains relaxing, so I would fly.

Fly from Naples.

Ditto to what Zoe and Suki have said.

We will have to check luggage but have been advised that its likely they'll be lost on in-country flights. Thoughts?

In 15+ years we have never had lost luggage on an Italian domestic flight.

We just took the day train/ferry from Napoli Centrale to Palermo Centrale yesterday. It was a long, but relaxing trip with great scenery, the chance to picnic on the train and get to know your fellow Italian travelers. We did not meet any other English speakers. It was a great experience.

Never lost a bag on a domestic flight, either.

Never lost a bag on a domestic flight. But if you are nervous about flying, you could take the Circumvesuviana from Sorrento to Naples and then take the train from Naples to Palermo. It's pretty inexpensive, and there is an overnight option (between 9 and 10 hours).
I took a train from Rome to Palermo once, and while it was long, it was worth when you are required to leave your seat and stand on deck while the train crosses the Messina Strait on a ferry. It's gorgeous!!!

"We will have to check luggage but have been advised that its likely they'll be lost on in-country flights."

Not sure where you got this "advice," but luggage is no more likely to be lost on a flight within Italy than anywhere else, meaning it's statistically rare (although obviously a pain when it does happen). This is particularly true since NAP to PMO is a short nonstop; it's not like changing planes, where there is a higher likelihood of your luggage not making the connection. Remember that when people have lost luggage or other issues, they can't wait to go online and complain about it; if they didn't lose their luggage, they don't post "we flew from Naples to Palermo and nothing unusual happened." So, problems seem more common than they actually are.

Multiple airlines fly this route; look at Skyscanner for options https://www.skyscanner.com/. Note that Alitalia has a more generous carryon allowance as well as one free checked bag, so it can work out cheaper than a "budget carrier." Go to each airline's website to do a dummy booking (up to the point where you put in your credit card) to check all rules and fees.

If you do want to fly Alitalia within Italy, be sure to use their "International" website: https://www.alitalia.com/en_en/. This has the same prices as the Italian website as well as an English option (the Italian website is in Italian only). The US website shows much higher prices - for the exact same flights!

If you don't want to fly, there is an overnight ferry from Naples to Palermo with two different companies: http://www.directferries.com/napoli_palermo_ferry.htm

And if you want to take the train, look at Trenitalia for routes and times. Even on this English website, use Napoli Centrale and Palermo Centrale for the station names. http://www.trenitalia.com/tcom-en

My preference would be flight over ferry, and ferry over train. But as always, the "best" way is in the eye of the traveler.

Do not under any circumstances take the night train. The ride was unbelivably jerky all night long. A miserable experience.

Thanks for everyone's advice — it was unbelievably helpful!