Travel from FCO to Siena with jet lag!

Enjoy your planning! Can I suggest looking at trenitalia.it for train schedules to your destinations. The Freccia trains make travel time much easier to take but it is still time. It will give you a better idea of how much time you will be spending in transit. Siena is beautiful and fairly easy to get to . . . I said 'fairly.' What I do (for what it's worth) is take the Freccia train from Roma Termini to Firenze and then the bus to Siena (it stops right outside the medieval city center). Timing? You're probably looking at 4 hours. The train is very comfortable. The bus ride is easy and scenic. You'll probably arrive close to check in time at your hotel or apartment.. There is a 'local' train from Rome to Siena. I don't recommend it for your first day = not always the most comfortable way to travel especially on your first day).
BTW, 4 nights in Siena is ok, I might suggest making it 3 and add time to Rome (have you been there before? 2 nights won't be nearly enough, I think).
It will take some travel time to get to Salerno. Siena to Florence (bus). Florence to Salerno will probably take at least 2 train rides to Rome then to Naples then to Salerno. Check schedules: is that taking a lot of your vacation time away from you? Again, considering the travel time, are 3 nights too many in Salerno? Of course, that decision is based on your interests and activities in Salerno. I do suggest a day trip to Paestum. The greek ruins are amazing.
Ending up in Rome, I think, is smart. It is easy to get to FCO even for a morning flight home. I wouldn't trust doing it from any place else.
Jet lag? Hey, you're in Italy! Get a good night's sleep the first night and I bet you'll be up and excited for the day in front of you!

I have the same qualms as your plan as Robert--I could not bear to lose that time in transit when there are so many places close to Siena and Rome to visit.
I find a train ride or bus ride on arrival is often the best way for me to get a nap in and deal with jetlag (I get little to no sleep on a plane), so I would press on to Siena upon arrival. You might be able to get a direct train to Florence, so then it would just be the bus ride, or you could arrange for a driver from Florence, since that would be less than all the way from Rome.

I would Take a cab from FCO to Tiburtina bus station and a direct bus to central Siena, on the top of the hill. The train station in Siena is central to the modern city, at the bottom of the hill.

For sure I wouldn't take a train headed north from Rome up to Florence only to take a bus going south and backtrack to Siena.

There is a train from FCO to Roma Tiburtina railway station, but you should walk from the tracks area to the bus depot. That's something I would avoid while jet-lagged, but staying on a bus for 3 hours and something shouldn't be a big deal.

Dario’s suggestion is the way to go.
There are several buses going to Siena from the Rome Bus station which is next to the Tiburtina train station. You could also take the train from FCO to Tiburtina station, but a taxi, albeit more expensive, is probably more comfortable and convenient if you have luggage,
There are several bus companies to choose from. Below are two of them, but there are more.

https://www.flixbus.com/bus-routes/bus-rome-siena
https://www.itabus.it/en/tratte-bus/rome-siena

Thank you so much for the input. I am in the beginning stages of putting this trip together so everyone's input is great. I think the cab to the bus station is the best way to go for us. Thank you so much for the links.

We are staying another 10 days in Italy after our friends go so we will be in Rome for a total of 4 nights. Then probably on to Naples and Bari.