I would not agree with the regionals they are mixed bag some are quite horrible, old and dirty. But the IC trains are also good.
We've caught a few Italian regionals in the last two months ... they've been pretty good to be honest. I can't claim they're all good.
Google Maps: trains, changing at Florence. Rome2rio: trains, changing at Rome. A direct flight from Genoa to Naples would certainly save a lot of time, though.

If you pick your trains carefully you can do the train trip from S. M Liguria to Naples in under 7 hours . I don't believe you could get to Genoa airport, fly to Naples airport, and then transfer into the city quicker than that. There's a departure around 1015 changing at Pisa and Florence and thus avoiding changing in Rome which arrives in Naples around 5pm -just check trenitalia.
A fair point from mike9. The flying option introduces additional links into the chain, whereas train travel is only one or two changes. Less hassle, and probably easier with luggage too. Trains are reliable.
The train ride takes 6 1/2 hrs, for example Rapallo dp 7.37 - Naples ar 14.00; change at Rome or Sta Margherita dp 15.15 - Naples ar 21.35, change at Chiavari and Rome
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