Are you spending a night in Florence at least? If you've never been, I highly recommend it!
As for using Cortona as your base, keep in mind that for day trips from Cortona, you'll use the bus and not the train to get to Siena and to Montepulciano.
Here's some info on the bus timetables using Google Maps:
https://goo.gl/maps/jTnuhCWV8MP2
and here for Siena:
https://goo.gl/maps/8pWZGwxt1JJ2
For Assisi, you'll need to do train and bus - here's a map with that info:
https://goo.gl/maps/tFHoW7z5ZjJ2
You can see most day trips will require careful planning - most are over 2-3 hours each way which doesn't leave a lot of time for sightseeing at each place. In the end, you might want to consider
renting a car to use while you're in Cortona as a base. The closest rental agencies are in Arezzo and Chiusi - you can do a search to see what comes up with rentalcars.com here:
https://www.discovertuscany.com/tuscany-car-rental/
If you don't want to drive while here at all, then consider using either Florence and Siena as your bases instead of Cortona.
You can actually go to
Cortona and Assisi from Florence with a guided tour, check this out:
http://tours.discovertuscany.com/en...ona-Day-Trip-from-Florence/d27667-6367ATOUR10
And once you're in Siena, a guided tour to Montepulciano and get to see more of the Val d'Orcia that way (adding Pienza in).
Check this tour out:
http://tours.discovertuscany.com/en...enza-and-Montepulciano-Wine-tour/d206-5919P28
From Siena, you can then continue on to Rome:
Flixbus now has direct buses down from Siena so you don't have to do longer trips by train (which generally takes you back to Florence or has you go to Chiusi and change trains there).