I am leaving for the US from Greece catching the connecting flight to the US in Italy, so where do I do passport control? My ticket says Athens via Rome to the USA.
Do I do it in Italy?
And when I return with my ticket USA via Rome to destination Athens -when then do I do passport control? Does it make a difference where I pick up my luggage? In this case where do I go through customs? Thanks for the info!

Earlier this year I flew to Italy via Paris. I went thru passport control in Paris but not again when I arrived in Italy. On the return I did not do passport control when I left Italy but I did in Paris. Just now I double-checked my passport my EU entry and exit stamps are both from Paris. I think you will do passport control in Rome both ways.

On the return you will pick up your luggage and go through customs at your first port on entry in the US.

This person isn't returning to the US. We have no idea what her residence or citizenship is.

Whatever happens will happen, and they'll tell you what to do. I don't ever worry about such issues ahead of the trip, what is the point. Sometimes it may even change from what someone tells you, so just follow the directions they tell you on the plane when you get to these places.

The flight from Greece to Italy is intra Schengen so no passport control. In Italy, you will exit Schengen and go through passport control as well as all the asinine checks imposed on flights to the USA by the paranoids in Washington (random checks at the gate yada yuada yada)....on the return, you enter Schengen in Italy and go through passport control...the flight from Italy to Greece again is intra Schengen...incidentally Schengen, as I understand it and could be wrong, only applies to immigration. You go through Greek customs upon arrival in Athens when you reclaim your baggage although when I've done an intra Schengen flight on a connection from the USA and picked up my baggage at my destination, there are signs directing me to customs for that ocuntry but generally I have not seen any Customs officials (not that as a visitor I usually have anything to declare anyway). I wonder, though, how that works for residents.

Passport control/"immigration": first and last Schengen airport, respectively. In your case: Rome (both ways)

"Customs" (in Athens): Is more a matter of seconds as there is no obligatory stop. You walk through the designated color-coded exit (red/green) after you claimed your luggage. Spot controls can occur, though. So don't use the "nothing to declare" exit unless you don't have anything to declare.

But: This is only true when your luggage gets checked through from the US to Athens.
If you have to, for whatever reason, claim your luggage in Rome, you will walk through "customs" there and check your luggage again landside at the counter for your flight Rome-Athens.

Outbound and inbound you go through passport control in Rome.

There's no outbound Customs. Inbound, assuming your bags are through-checked to Athens, they are officially liable to Customs inspection on arrival in Athens (the baggage tag signals to Greek officials your bag originated outside the EU). In practice, there's almost never any inspection, and if there's something about you or your bags that causes an alarm, you might be searched in Rome.

Your residence and citizenship are totally immaterial.

Have your passport handy in Athens, because the airline agent checking you in may want to see it, since you are through-ticketed to the US and the airline that gets you there has to take you back if your entry status is not satisfactory on arrival. Same again in Rome of course.