I presume you meant to put this Q in the forum for the Region of Tuscany - https:/…ShowForum-g187893-i67-Tuscany.html (Toscana in Italian), of which the capital is Florence; rather than in the forum for the town of Tuscania in the Region of Lazio, of which the capital is Rome!
But in any case, Arezzo is very close to the border with Umbria, and some of the day trips you could make would be to that Region (e.g. Perugia, Assisi and Spello). It's actually a terrific base for exploring by public transport.
From Arezzo's station there are direct R trains (no booking required or possible, and the price is the same whenever you buy tickets) to e.g. Florence (34 mins - 1h21 depending on departure), Terentola-Cortona (25 mins), Chiusi-Chianciano Terme (33-46 mins), Perugia (1h06), Assisi (1 1/2 hrs), and Spello (10 mins on from Assisi).
By faster Frecce trains (booking required and prices are lower booking up to 3 months in advance) and/or changing in e.g. Florence, you can get to Orvieto (1 hour) and Rome (2h10) in one direction and Bologna (2 hrs) and Ferrara (2-3 hrs) in the other.
You can also get to Montepulciano in up to 2 hrs and Siena in 2 1/2 - 3 hrs with a change in Chiusi-Chianciano Terme, although it would be better to take the bus to Siena as this brings you to Piazza Gramsci within the city walls, whereas the station is outside the city.
There are buses to e.g. Sansepolcro (bus 381 to Città di Castello permits a day in Sansepolcro but only 1/2 day in CdC) Siena (138, 1h30: http://www.tiemmespa.it/var/ezdemo_site/storage/original/application/d4403b171514287d8bfceb8f42cecb81.pdf) and Florence (2h35) (full "extra-urban" bus timetables for Arezzo area: tiemmespa.it/index.php/…Extraurbano - click on "mostra altri" at the bottom to continue).
I leave it to you to research what there is to do in any of these places that might interest you, but for example Sansepolcro is the birthplace of Piero della Francesca and its Civic Museum is worth visiting if you are interested in art (…intoscana.it/allthingstuscany/…).