Rail from Rome to Cortona

The question is really "does Cortona have a station?" (I know Rome does!).
I went to the Wikipedia page for Cortona ( https://en.wikipedia.org/wiki/Cortona ), and at the bottom it says:

Transportation
Cortona may be accessed by rail: the closest station is Camucia-Cortona, 3 kilometres (1.9 miles) away. There are direct trains from Florence, Rome, and Foligno (via Perugia).

So I went to the Trenitalia site ( http://www.trenitalia.com/tcom-en ), and looked up trains Roma ( Tutte Le Stazioni ) to Camucia-Cortona for a random date, and I see lots of trains, taking around 2½ hours, prices starting at €11.90.

That is the way you find out. Very easy.

... or use that wonderful invention: a map.

What Ton said, and as he did use OpenStreetMap.
Google Maps is useless for train travel, it doesn't show rail lines or stations unless you get very close.

I took the train from Orvieto to Camucia-Cortona and then the bus up to Cortona. It was a train that had originated in Rome, and the trip was very easy. Buy your bus tickets at the newsstand in the railway station or, if closed, at the store on the left about a block up the street leading away from the train station.

For easy starting research into how to get from one place to another I like to use the Rome2rio website. The driving times are not very accurate and prices are random but for basic 'here to there' options it works well. Then you can go from there to specific transportation websites to get details.