Greece and Sicily?

With 10 days in Greece I would say you basically have 3 possible options:

1 / Athens and 2 islands maximum

2/ Athens, a nearby island, and a small part of the Peloponnese

3/ Athens and a larger part of the Peloponnese

And since you want to plan the trip yourself, it's up to you to find out by consulting travel guides and especially by looking at a map.

You need more time in Greece unless Athens and one island is enough for you. Or Athens and Nafplion. Not all of them.

You need more time in Greece unless Athens and one island is enough for you. I would feel very cheated with just those.

And if this is likely to be your only trip to Sicily, I'd want at least 2 weeks there. I would definitely not split a 2-week trip between Sicily and Greece.

Check your flight options between those two destinations. They may be limited without connections, which can chew up a lot of time and mean costly airfare. And watch out for unprotected connections.

Everything depends on if it's two or three weeks, and if there is a direct flight at the time you visit (some flights are seasonal). Two weeks would not be enough time for both. Three weeks gives you the options listed above: a limited visit to Sicily (maybe you are okay with that) and a two-destination trip in Greece. I visited the Peloponnese over ten days (three bases) with a limited stop in Athens, but I am sure I'll be back to Athens, so I was not worried about giving it short shrift.
Whether to chose islands or mainland in Greece would depends on personal interest but also what you'll be doing in Sicily (which has some gorgeous islands as well). Don't forget the mainland is ringed with beautiful seas and beaches--I found it maximized my time much more to drive the mainland versus flying and ferries.

If history and architecture are of interest: noting that on our Sicily tour, we heard several times, that there are more Greek ruins in Sicily than in Greece.

Thank you for your replies. Lot's to consider. I may have to plan two separate trips - one to Greece and one to Siciliy.

That's what I would do!
As for May or September, May is really lovely--my trips to Italy in late May have usually been perfect (except this year), and my one trip to Greece in late May was similar--great weather, low crowds. I have always been tempted by September though, for the warmer seas. But since time of year could possibly affect direct flights, I'd check that first.

Days are a lot longer in May than in September, but water temp will be a lot cooler. I don't know about rainfall.

It might be getting quite hot in late May and will likely still be hot in early September.

I have been in this situation countless times. In recent years, perhaps due to age but not necessarily, I have taken to closing the circle somewhat, as it were.

Namely, here, instead of flying 600 miles to Athens, how about getting in a car and either diving deeper into the more-distant reaches of Sicily? Or, across the channel to check out Calabria, Puglia, Basilicata? Either will allow you to leave the vast number of tourists behind and you'll gain a much greater appreciation for the place you came to see.

Greece will still be there when you are ready to visit Greece.

My two cents; enjoy your trip.