Greece, Croatia and Italy June 2020

Personally if it was me taking my three kids of that age by myself I would totally do a cruise. There are a ton of cruises that hit the exact itineraries you mention; most are about 12 to 14 days long. You could tack on extra days at the beginning or the end as many of the cruises start in Venice, Rome, or Athens.

Not sure where you are located but if you have a free place to stay in New York you definitely may find cheaper airfare from JFK then other areas of the US.

You could start in Greece. They now have flights from Athens to Dubrovnik. Then I would make my way up the coast of Croatia, take a catamaran from Rovinj to Venice. I would then take a few days in a few of the smaller Italian cities, then Florence if possible, then fly home from Rome. The possibilities are varied and plentiful. This is the route I would plan and believe me, just like many posters here, I have about 10 routes roughly planned out for future trips.

You can easily spend a month in each of the countries you have mentioned, so you need to work out what you want to see in each country and get your children involved in this process. Why have they picked these 3 countries?

From Greece, you will need to fly, so it doesn’t matter if you fly to Italy or Croatia next, as you may find better flight connections into one place or another. The flight times won’t be much different and most of the time will be getting to and from the airport and waiting for the flight.

Between Italy and Croatia, you can fly or take a boat, although this depends on where in each country you want to visit. Fly open jaw to avoid back tracking. You also need to consider which places are easiest to fly in and out of for your transatlantic flights which may influence the order of places. Check Skyscanner for flights. Rome2rio.com maybe useful for itinerary route planning.

You see relatively little of each country when on a cruise, so I wouldn’t recommend this option. Are you prepared to hire a car in each country? How much beach time do you want? Late June will be hot in all three places.

I will start with Italy, since it is my favorite European country to visit. It has so much history, art and culture. Also, great food.

Rome, Florence and Venice are the big three and you need two weeks minimum for those cities. Perhaps you could add Sorrento, Capri and Pompeii for 3 days, if you have time.

I suggest flying into Rome (5 -6 days there) then moving up to Florence (3-4 days) then on to Venice.

From Venice, I would find a cruise that stops in Split and Dubrovnik, Croatia. Those cities are not easy to get to my driving. Also, you can see them in one day (do a side trip to Trogir when in Split).

Hopefully, your cruise will go on to the Greek Islands and includes Corfu, Athens, Mykonos and Santorini. Plan to fly out of Athens (open jaw ticket).

You need 3-4 days in Athens to take in all the important places as well as the New Acropolis Museum and the Archaeological Museum. Also, consider a day trip to Delphi.

Have your kids read up on the places you plan to visit, so they have some knowledge of the history. They don't need to read , The Peloponnesian War, but some guidebooks or
https://theconversation.com/guide-to-the-classics-thucydidess-history-of-the-peloponnesian-war-71550

I have traveled to Europe with three kids that age several times (my kids are now 17, almost 15, almost 12), although not to Croatia. Its true that each country would be great for a month, but since that is not your desire, take the "you need 5-6 days" in this place or that with a grain of salt. I would definitely talk to your kids about why they are attracted to these places and what they want to do. We went to Italy for 2 ½ weeks and never went to Venice (we were in the Cinque Terre, Florence and Rome), and spent 4 ½ days in Rome and found that absolutely plenty (geovagriffith may have had side day-trips in mind within that 5-6 days however). We went to Greece for a month and missed several places that many people recommend. The point is, what is interesting to me and my kids won't be the same as everyone else, and the pace you like to travel won't be either, so think about that. I find 4-5 full days in big cities (Paris, Rome) about right for us, and anywhere from 2-4 full days in smaller areas, depending on how many day trips we plan (we spent 2 ½ days in Florence without day trips, for example, but 4 full days in Provence with many day trips).
As for logistics, I would look at flights in and out of LA (which looks like where you are from). Are you familiar with Google flights? Its a useful algorithmic flight finder where you can play around with dates and departure and arrival destinations to find the best flights for you in terms of price and time. The route to Europe from LA isn't necessarily through the east coast, so you may end up spending more money and time getting to JFK first. Because we are budget travelers, I play around with departure airports (that I could travel to from my city relatively cheaply, either on a separate flight or driving) to find the best price. Most years I have flown out of one airport, which routed us through JFK, another year, the best price was to fly out of LA, and that flight went through D.C. Point being, I dont know that your mom's location will really matter, and I suspect that getting to NY on a separate ticket will make the whole thing more expensive and longer, but again, play around on Google flights (I like it for the "Date Grid", where I can see the difference in prices from day to day over a month), but some people like Skyscanner. And consider the "multi-city" option (different arrival and departure airports?).
I like the idea of flying into Greece and out of Rome. Of course you could go the other way too.

I suggested starting in Greece because i like to start at the further point and work my way back. The longer flight time or multiple flights are always easier when you are starting your trip and you are excited and fairly well rested. By the end of the trip, you are tired and just want to get home as fast as possible.