Hi Heather,
You can find many information about trains in Italy in the
official site for Trenitalia, Italy's train operator. Did you already buy a pass for traveling in Europe or Italy by train? In this case, I suppose you have some restrictions which you should already know of.
If you don't have a special pass, you can travel very easily by train throughout Italy. From Rome to Florence, the fastest trains (more expensive of course) are called Eurostar trains. These trains generally need a reservation but you can still buy tickets up to a few minutes before departure if there is space available. You can also buy tickets online through Trenitalia's website.
There are no direct trains between Rome and Siena, however. You'll have to change trains at least once.
From Florence, you can reach Siena either by train (change in Empoli) or by bus. The SITA bus company offers direct connections, you can view the timetable here
http://www.sitabus.it/sita-toscana/Firenze%20-%20Siena.pdf. You can buy the tickets at the bus departure area located in Via S. Caterina da Siena next to the Santa Maria Novella train station in Florence.
Vagliagli is about 15 km from Siena. I found that the bus 34/A connects Siena to Vagliagli but looking at the timetable
http://www.trainspa.it/URBANO_SIENA/034A.pdf it seems that there are no buses to Vagliagli on Sunday. I suggest you contact the Villa directly for information, see whether they might offer some suggestions or organize your transfer from Siena to their accommodation.
I hope my suggestions will help!
Have a nice trip to Italy, I'm sure you' ll have a lot of fun!