We would like to travel by train from Florence to Athens Greece in one day.

Have you looked at a map? Fly, too far for a one day train.

The only train that would be involved would be from Florence to FCO, to get a flight to ATH

I wonder if there is a continuous rail track over the itinerary, let alone a train running over it. I think not.

Unfortunately that's not possible as there are no rail connections, especially in one day! Your best options are....

• train to Bari and then take a Ferry to Patras and then bus or whatever to Athens.
• train to Milan and then take one of the non-stop flights using Aegean, RyanAir or Alitalia (that would be the quickest and easiest option).
• train to Rome and take a non-stop flight from there.

Fly. Cheap and easy from Rome, or Milan, probably a connection from Florence (not sure).
Or if you really want to avoid planes then take a ferry from southern Italy to Greece, then bus to Athens.

Look at a map. A train would need to curve around the top of Italy, go down through Slovenia into Croatia, Bosnia, then Albania before crossing most of Greece. Meanwhile a flight just has to cross the Adriatic. I also don't think trains are very extensive in all parts of the Balkans, especially Albania.
And Greece itself isn't a big train country. It has some trains, but not nearly as many as, say, Italy. Travel through Greece is generally by car, boat, or bus.

I tend to reference this website a lot, but rome2rio dot com lets you see travel options between locations. Also lets you try variations of multi stops to see how it plots for backtracking or cities that are outliers.
I don't rely on them for current price and length of time, but it is often my first level of research

This is the first post by the OP.

OP: A good start would be to get some guide books and maps, and figure out some distances before you book anything.

Check out the oracle for all things rail travel, The Man in Seat 61. He describes a rail route from London to Athens, going via the length of Italy, and taking a ferry from Bari to Greece. This route could be adapted to start from Florence. But I doubt it could be done in one day.

Route description here: https://www.seat61.com/Greece.htm#Bari%20&%20Patras

This is a duplicate post with more answers over there.