Mary,
First of all, about the ticket resellers. I've used Trainline a number of times in the past for travel in many countries in Europe and that's always worked well. I don't mind paying a small surcharge but that's an individual choice.
If you wish to purchase tickets from Trenitalia, that will be the cheapest option as they're the company that operates the trains. I checked tonight and found many departures from Siena to Rome on October 10th, using various trains and with different prices. One of the most cost effective trips will be a departure at 08:13, arriving Roma Termini at 12:48 (time 4H:35M, one change in Chiusi with a layover of 1H:18M). The first train is a Regionale and the second train is a Regionale Veloci (a bit faster), and both are Second Class seating only. The price is shown at 36.20€ (I don't know if that's for one or two people). If you want a faster trip, shorter transfer time or a different departure time, you'll likely pay more. There are all kinds of options. Neither of those trains has reserved seating, so you'll just have to find the first open seats.
The Regionale trains are "no frills" but they will get you where you want to go. You can easily buy the tickets for those locally, since there are no discounts for advance purchase. However if you do that, you MUST validate (time & date stamp) in the machine in the station or risk hefty fines collected on the spot. If you buy the tickets online, I believe they're pre-validated but can only be used on the date listed on the ticket.
One other option would be to travel via Regionale from Siena to Florence, and then connect with a Freccia fast train. Of course the cost will be higher but the trip will be quick and you'll have much more comfortable seating in the Freccia. From Florence you could also use Italo Treno. If you use that option, your departure station will be Firenze S.M. Novella and your arrival station will be Roma Termini.
You might find this video helpful. It's a bit dated but still has some good information - https://www.youtube.com/watch?v=6n8rITO1Eek .
As the rail station is at the bottom of the hill below Siena, you might want to ask your hotel staff to pre-arrange a taxi to get you to the station in the morning.