Just agreeing with the others. Your destinations - Athens, other parts of Greece, Sicily, and Capri - do not have good transit links between them. As acraven detailed, many of these pairs required involved connections, such as two planes, or a plane followed by a bus ride of a few hours. These not only take more time, but are more tiring; it's hard to "hit the ground running" when you've spent much of the day just getting there.
Even though 20 days sounds like a lot, you may have to pare back your wish list to have an enjoyable trip.
To find who flies where from a particular airport, look at that airport's Wikipedia page. Note that when a flight is "seasonal," the season can be long or short. Flights are also not always daily; they can be as infrequent as once a week.
To find specific flight schedules and where to book them, use Skyscanner: https://www.skyscanner.com/
To investigate connections between any two points, look at Rome2Rio: https://www.rome2rio.com/ However, you should never take this as the last word. From Rome2Rio, get the name of the specific airline, bus company, etc, then check their website directly,. particularly for prices and schedules. But It is good for identifying snags you didn't otherwise know about.
Depending on where you are in the US, even flying back from southern Italy can be complicated. You may need to get to Naples, or even Rome, the night before a flight to the US. Investigate this before finalizing any plans. One that often comes up is people wanting to be in Sorrento, Positano, or Amalfi the night before an early flight out of Naples. This requires a pre-booked taxi costing €100-200 (depending on distance), not to mention getting up at oh-dark-thirty to catch this taxi.
To find transatlantic flight options, use Kayak: https://www.kayak.com/flights or Google Flights. Be sure to use the multi-city option (for instance, into Athens and out of Naples).