Train transportation from Rome to Sorrento in June

You can reserve train tickets from Roma Termini station to Napoli Centrale with an advance discount up to 4 months ahead at www.trenitalia.com, or buy them on shorter notice as you prefer (full price around $50 per person in 2nd class). From Naples to Sorrento, buy tickets in the station for the local Circumvesuviana commuter train (running twice per hour) or the less frequent, tourist-oriented Campania Express. Rick's Italy guidebook has good details in the relevant chapters.

Departing from Roma Trastevere or Roma S Pietro stations would not really help you, as it sounds like you're not staying very close to them and you'd still have transfer to the long distance train at another station.

That's easy. Freccia (high speed train from Rome to Naples (Centrale stn). Then the commuter train (Circumvesuviana ) to Sorrento. That train is downstairs from the main Naples station. Buy that ticket before you board. Slightly more expensive than the Circumvesuviana, but faster and more comfortable, is the Campania Express.

Look for the train schedules and prices for the Rome-Naples trains on the train website. ( You need to use proper Italian place names: Roma, Napoli) https://www.trenitalia.com/en.html If you have problems with your credit card on that site, try buying them from Trainline.

You could also take an Italotreno high speed train to Napoli Afragola station and then a bus run by the same company to Sorrento.
Now it's really too early for train tickets, in April look for tickets on www.italotreno.it/en and simply pick Sorrento as arrival station.

We did exactly that trip last June. I just followed the advice in RS Italy guidebook and it was easy. As was mentioned above, if you buy your tickets further ahead you will get a discount (and ensure your train is not sold out). I do strongly encourage you to buy ahead, as we ran into trains being sold out in 2nd class last summer. By early, I mean several weeks ahead of time not months before your journey.

We flew into FCO and were going directly to Sorrento, but our flight was 1.5 hours early due to a strong tailwind (this was a first, the flight is usually very late). I never make frecca (reserved seats on the fast train) the same time as expected arrivals, so instead I booked tickets on Fiumicino express bus that left from the same terminal after customs. The bus was very nice, not crowded and very clean. We were the only non Italians on the bus and the only issue was no bathroom (which is good and bad, but I wish I had known). The bus went directly from FCO to Napoli Centrale, as the bus lets you off at the train station. It saved us from going to the airport, then Termini and getting on a fast or regional train to NC. When we arrived, we went downstairs to the Circumvesuviana train to Sorrento. That train was not so nice (a group of men got on and started pushing and shoving and then jumped off) and then people got on and started singing and playing loud instruments and then put their hat out for offerings. But it was a quick ride to Sorrento and worth the effort.