Paris to Rome to Athens
 

I have a trip planned this summer to go to Paris then Rome then to Athens and Santorini. I wanted to get advice on traveling between those cities via rail vs. plane. How can I incorporate other cities along the way? I am going from August 19 to September 1.

With the number of days you have, forget about other cities and fly between the ones you have already planned. Paris to Rome can be a very interesting train trip, with some good intermediate stops (Lyon, Switzerland, Milano, ...) but you don't have the time for that if you also want to visit Athens and Santorini.

Paris to Rome, via train, will almost an entire day in a train, even if in a high-speed one.

Rome to Athens means several hours in a non high-speed train then 12-20 hours on a boat (depending on which port to use) and then another 3-4 hours on a bus (no more trains from Patras to Athens for another X number of years).

With 10 days available, planes are the only alternative. Mind you, these three cities in just 10 days? I personally doubt that you'll enjoy your trip!

Looked up Paris-Rome: two possibilities to do it in ~13 hours, both leaving early in the morning or, alternatively, a few including the night and being in the train for >18 hours!

I certainly think that flying is the only sensible way from Rome to Athens but I do think overnight train from Paris to Rome is a possibility. I agree that you haven't really got the time to make the journey by day train unless you really want to.