Even pre-Covid, the links between the various Mediterranean islands were skimpier than one would assume. Flights often required connections. There were very frequent flights from Rome to Sicily and Rome to Sardinia, but flights to other places were sparser and in some cases non-existent. And connections between the islands that did not involve returning to the mainland were minimal. I have no idea what's available now, and as said above, flights and boats are likely to increase as time goes on. Furthermore, not all flights are daily, so you have to check your exact dates.
One thing to know when searching for flights to or from Sicily is that while Palermo is the bigger city, Catania is the bigger airport with more flights. So, there may be a flight from Catania to Sardinia on a day when there isn't one from Palermo. And, Sardinia has 3 airports - Cagliari, Olbia, and Alghero. I forget which one is busiest, but do check all three.
Genoa airport is small and even pre-Covid had relatively few options. To fly there, you're likely to have to transfer at a hub such as Rome, Munich, etc. Or, you can fly to Milan and take a train or bus from the airport to Genoa.
For Sicily to Naples, you can fly, take a long train ride (overnight or by day), or take an overnight boat (from Palermo). The boat gets relatively good reviews. The overnight train does not, although some don't mind it. I'd fly, but if you want to take the overnight boat, see if it's restarted running for your travel dates.
If you will have to change planes, do make sure you don't have to change airports. Any potential cost saving will be negated by the money, time, hassle and stress involved in getting from one airport to another. This is a particular issue with Milan, but can come up with other cities including Rome.