According to what I have read there are 2 choices: a bus can take you from Ciampino to Anagnina metro nearby and then you get the underground or else there are more infrequent and expensive direct buses there.
Which is better?
There are multiple private buses there, for 4 euro p.p. to termini. 1 hour trip about...
Alot of tourists get robbed on the underground and public buses apparently, one route is referred to as 'the wallet eater'.