Transportation options from Porto Santo Stafono to Rome

A good place to see your travel options is the website https://www.rome2rio.com/ Shows trains, planes, busses, rental car drive time. A search shows trains take about 2:15 and leave about every other hour.

Don't trust Rome2Rio for details like travel time, frequency or cost. It can be wildly inaccurate. Dig down to find the name of the rail or bus company (there's often a link) and go to its website for accurate information.

From Porto Santo Stefano you would take the bus operated by AUTOLINEE TOSCANE to the train station called ORBETELLO-MONTE ARGENTARIO

The bus line number is Oo1
Oo1 Bus Timetable here also downloadable in PDF format for both summer (estate) or winter (inverno)
The bus ride is about 22 minutes.

Once at the train station you would take a train down to Rome.
Train Timetables and prices available on the Trenitalia website
Enter from: ORBETELLO-MONTE ARGENTARIO
To: ROMA TERMINI
Select date and time of travel.
If traveling far ahead in the future just check any date in the next week
The train ride is less than 2 hours.

We are staying at the Hotel Tiber which is very close to FCO and need to travel by train to Porto Santo Stafono. Should be go back tot he airport to get the train or go to the train station in Rome. We will also be taking the same route on the return about a week later (Porto back to Hotel Tiber). Do we need to make reservations and purchase our tickets in advance?
Thanks appreciate the guidance.

The hotel Tiber is in the town of Fiumicino, which no longer has a train station. You will need to go to one of the stations in Rome to take the train to Porto Santo Stefano (the closest train station is ORBETELLO-MONTE ARGENTARIO, from which you take the Oo1 bus to Porto S. Stefano).

The best option is probably to go back to the airport (by taxi or hotel shuttle) then depart from the Fiumicino Airport train station. A change will be necessary at one of the stations in Rome. Which Rome station you change trains depends on the train departure you choose. For faster trains you will need to go to Termini (Rome's main station) but for Regional Trains or some Intercity trains you can use either Trastevere or Ostiense stations.

go to https://www.trenitalia.com/en.html
then select:
From: FIUMICINO AEROPORTO
To: ORBETELLO-MONTE ARGENTARIO.
Enter date and estimated time of departure.
You will get a lot of trains to choose from throughout the date. Check the ones that interest you.
You don't need to purchase in advance, you can buy the ticket at the airport station the morning you go.
If you choose a fast train (like a FrecciaBianca or IC-Intercity) buying in advance offers the chance to save because of discounted fares which tend to disappear fast. However if you buy a ticket in advance, then you are locked with that choice and if you change your mind you might lose your investment as discounted fare tickets are non refundable or changeable, except with heavy penalties.
If you choose to travel on a cheap Regional train (a bit slower as it makes more stops), there is no need to buy in advance, since those trains do not offer discounts.
If you buy a ticket a the station, don't forget to validate your paper ticket (date/time stamp) at the validating machines before you board the train. Digital tickets purchased online do not need validation.
Once at ORBETELLO-MONTE ARGENTARIO station, you need to take the Oo1 bus as explained in my previous post to the OP.
To return to Fiumicino, you do the reverse.