Fly into Rome and back from Athens, or vice versa?

For me, Athens was easier to navigate than Rome. It's probable your total travel time would be longer into Athens versus Rome as your first city. Also, Rome is one less hour time difference, so jet lag would be slightly less. With those two considerations, Rome wins.

Alitalia fly direct to Rome from Chicago.
Book multi city return, connecting in Rome for the homeward journey from Athens.
As far as I can see no carrier flies Chicago - Athens direct.
Spanish low cost carrier (Iberia subsidiary) offers low fares Rome-Athens if you need to make your own arrangements between the two cities. I don't think it matters where you head first - I would see which option offers the most convenient arrival/departure times.

You indicate three weeks in Oct and perhaps a Greek Island. I would start in Athens since the island begin to shut around after the first of Oct. Airline schedule between the islands and Athens change as does the boat schedule. So I would plan the Greek activities related to the island as early in Oct as possible.

I suppose Athens because of its smaller size is easier to get around but Rome is equally easy. I personally find Rome very easy to get around but since we have spent nearly a month in Rome over the years we have a lot of practice.

Finally, later you get into Oct the better the weather is in Rome - cooler and should be dryer. Our preferred time in Rome is always around mid October. And take a look at booking your airfare not just as an open jaw ticket but as a three legged flight. Probably very reasonable.

I like to get my longest flight out of the way first. I would also love a non-stop on the way back.

I am with you Zoe. I think i'd head to Athens first (perhaps via Rome connection) and I also think a Greek Island would be better at the beginning of October.