In addition to Simone Carletti's answer, there is also the "Bus Shuttle" operating between Ciampino Airport and Termini (see
http://www.sitbusshuttle.com/en/ciampino/). It operates as a competitor to Terravision on that route.
In Ciampino airport, their ticket booth is adjacent to the one of TerraVision. Since you are asking for recommendations, I would suggest to buy a ticket to Termini from one of these booths, for whatever bus is the next one. With a ticket price of 4 Euros (one way) for any of the options, splitting up the trip into a bus and a train part is likely to be not worth the hassle.
For the trip back, you need to buy a ticket ahead of time (e.g., on TerraVision's website). Otherwise you will pay the walk-up fare of 6 Euros. You will also pay this fare if you do not visit the ticket booths at Ciampino airport, but rather walk to the bus straight.