Travel from Sorrento to Sicily.

The best way is to fly from the closest airport in Naples, which is a local train ride away from Sorrento.

I agree. Flying would be best into Catania or Palermo.

Fly to Catania.

I'd choose my Sicilian airport based on what places I wanted to see in Sicily. If it's a long-enough trip to cover both sides of the island, it's often best to fly into one of the airports (the two major ones are Catania and Palermo) and out of the other.

If you're staying in Sorrento, which provides the most transportation options, Pompeii Scavi, Amalfi Coast and Capri, use the Curreri Viaggi bus to Naples Capodicino Airport. (NAP)
You can reach both Sicily airports from there.

I agree with the above points:

1) Flying is easiest.
2) You can fly into Palermo or Catania; choose based on where in Sicily you want to go. If you have enough time to see several parst of the island, you can fly into one and out of the other, which saves time by avoiding backtracking.
3) If you are staying in Sorrento, you can take the Curreri Viaggi bus right from Sorrento to the Naples airport; this is easier than taking the Circumvesuviana from Sorrento to Naples, then getting from the city of Naples to the Naples airport as a second step. The bus runs every two hours or so.

Some additional tips:

To find flights, use Skyscanner: https://www.skyscanner.com/.

Some of the airlines that fly these routes are budget carriers, meaning that carry-on baggage limits are low and checked bags cost extra. So, do a dummy booking directly on the airline's website to confirm all costs. Alitalia includes one free checked bag as well as a more generous carry-on allowance, so it can work out cheaper in the end.

For these flights within Italy, the International website of Alitalia https://www.alitalia.com/en_en/ often has better prices than the US website. The Italian website has the same prices as the International one, but does not have an English option, which the International one does. Compare all costs before buying, and remember that buying from the non-US websites means a foreign transaction, so be sure to notify your card issuer first, and be OK with any fees charged for foreign transactions.

The cheapest tickets on these routes can sell out way in advance, and the prices only go up closer to travel. So, book these flights as soon as you are sure of your plans.

If you don't want to fly, there is an overnight ferry from Naples to Palermo that gets good reviews. The long train ride from Naples to Sicily does NOT get good reviews, and is only recommended if you're stopping off the see places along the way.

Addendum: I just looked at Alitalia's website, and I see that they have changed since I last flew them to Sicily in 2014. The cheapest fares are now "economy light" and do NOT include a checked bag. Understand all the rules and restrictions of any ticket you're considering, before you buy it.