Italy and Greece

Different tickets means that if there is a delay causing you to miss your second flight, you have little recourse. It's also an awful lot for 9 days--look at the logistics day by day to see if it would work for you, but I recommend one or the other.

Yes its possible to catch separate flights, but with risk, as noted above (Ive done it, but it was stressful hoping my first flight wouldn't get delayed, and a big relief when it worked). And I agree that your plans sound very full for 9 days. Can you list your itinerary with days in mind for each?

As the others have suggested, you are trying to do too much in the time you have. Nine days on the ground (really eight, because you are likely to be sleep-deprived and jetlagged on your arrival day) is actually very short for just Greece or just Italy. You will have a far better vacation if you choose one of those countries. Adding a few more days would help, too.

This is way too much for nine days! And focusing on Santorini over other islands is not a great idea either.
Either one of your destinations, Greece or So. Italy, will be rushed with only nine days.

How many nights in the ground?
Three nights are two full days, two nights are one full day.
Athens will need 3 nights (2 full days)
Mykonos and Santorini a minimum of two nights each, i.e. a full day, (although Santorini probably deserves 3 if you also want to see Oia).
I’ve used 7 (or 8) nights already. How many are left for Naples area. Four nights should be the minimum.

Way too much for the time you have. You are severely underestimating the time you’ll need to travel and not giving any of these places enough time to enjoy. Pick a country, but not both, unless you have at least 2 weeks or more on the ground.

With 9 days, unless you take a cruise, pick one country.

The Greek Islands are best seen with a cruise.

You will need all nine night for Athens, Mykonos and Santorini. Pick either Greece or Italy. If Greece, fly into Athens and fly to Santorini 3/4 nights so 2/3 full days ferry to Mykonos. Mykonos has some beautiful beaches 2/3 nights then fly from Mykonos to Athens. Or Fly into Naples spend a couple nights there then off to Sorrento and the Amalfi coast. You could end in Rome and fly home from there. And I disagree with a cruise for the Greek Islands. It gives you only a couple of hours in each island.

I disagree that a cruise is the best way to see the Greek islands in general. However, if you have a long list of islands you must see and very little time, a cruise may well be the only practical solution. Just remember that you may not have time to see much more than the port town on each island, and you will be accompanied by all your fellow cruisers.

We are thinking Athens, Mykonos, and Santorini. We would fly into athens or santorini and then ferry or fly between? Or maybe just see Athens and Santorini? Then, flying back into Naples where we begin our South Italy tour of Pompeii, Capri, Sorrento with some time down in Positano/Amalfi area. Is this doable in 9 days?

No

"Pick one" is what you need to do, to avoid a miserable experience

I'll add my "Don't Do This" to the rest of the responses. Island-hopping in the Cyclades, while it sounds romantic, will basically eat up a day waiting to board, waiting to depart, slow-going and a day mostly gone in transport. I'm not certain about the flying between islands. Most of the carriers I am aware of just go from the departure island to Athens. Then from Athens to whichever other island you want to go - no direct island-to-island. But I could be wrong here.

However, if you do insist to ferry between islands, while SeaJet will be quicker you will be enclosed inside the ship. There are no decks to wander, if memory serves.