Greece and Italy

Sicily is worth 2 or 3 weeks by itself, so take care that you don't have too high a ratio of travel time compared to sightseeing time. You'll need to visit just a small part of Sicily to make your proposed schedule work. Capri is also time-consuming to get to.

Only Athens has non-stop flights to Naples and Catania (Sicily); there are no non-stop flights to Palermo. This means you'll need to double back to Athens as you head to Italy.

Travel around Greece is rather slow. The mainland has very, very few trains, and they are not fast. If you want to see one or two islands, you'll need to take ferries (not fast) or fly--and flights may not be frequent. If I had 20 days, I'd use all of it in Greece. Not because I don't like Italy, but because Greece is often rather expensive to reach, and I'd want to have enough time to get a good look at it. Ten days wouldn't be enough time for me.

Until the Greece part is fleshed out it is difficult to advise, but acraven makes the strong point about not overdoing it. I do think a Greece-Italy trip is possible, but you have to be very careful. You'll also need to get specific about Italy because you can only see a bit of Sicily in 10 days.
If you are talking about the blue grotto on Capri, be aware that in late October it might be a bust. In early October, I would not hesitate to go to the Amalfi coast area, but as you get deeper into October the chance of bad weather increases.
If I was going to stick with Greece, I'd drive the Peloponnese and then fly to Crete--just have a hunch the weather would stay stable there longer. If doing Greece plus southern Italy, I think I'd save Sicily and spend about a week in the Naples/Amalfi coast area, giving the balance of the days back to Greece.

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.

Some tips:

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).

As the others mention, it depends on what you want to see in Greece.

One thing I once did when I visited both countries in a single trip was to fly into Athens, rent a car one way for sightseeing through the mainland, then flew from Corfu to Naples. After traveling around in Italy, I flew home from Rome. I imagine if you are interested in visiting Corfu, you could use it as your access to Italy on a direct flight or a ferry, then return to the U.S. from there. With 10 days, you would need to decide which key places you want to see most so that you leave enough time for enjoyment, but it is definitely do-able.