Travel between cities by bus and / or train

You can find information using Rome2Rio, but then I would look at the individual bus companies. Same with Omio. There are lots of buses from the Tiburtina Bus Terminal to Siena. I'm pretty sure you can buy when you arrive, if you fly in that day.
Same with trains. The main company is www.trenitalia.com, but there is also the private company www.italotreno.it, which operates only high speed between major cities (but not Siena). You can buy ahead and possibly save money, but if you fly in the same day then I would buy on the spot when I arrive and pay full price. Discounted tickets purchased in advance have restrictions (no refunds or changes) therefore if you miss the train because of a flight delay you lose your money. Siena from Rome is probably best by bus, otherwise you have to go to Florence and backtrack from Florence to Siena by regional train or bus

For longer-haul routes like Rome to Siena the schedules are likely pretty regular but this isn't always the case if you are also looking at regional bus routes. These routes, such as within Tuscany, often are aimed at getting people to/from work and school. So they may have special schedules or not run during vacations, on holidays, and so on.

All good information here so far. Fast trains can be booked ahead of time at a - sometimes serious - discount by trading flexibility for price. Pretty much all other trains like the local regionale trains and buses do not need to be booked ahead of time and can be booked as and when needed.

Get the Trenitalia app to check schedules and book trains when you need them. I would be familiar with any train routes you want to take and have an idea of the schedules and frequency on the days you want to travel but you don't need to buy ahead of time.

Rome2Rio is a great resource for general checking of travel times and connections but always verify the data they are showing you and don't use it to make firm plans based solely on their results.

Have a great trip,
=Tod