From Rome Fiumicino Airport (FCO) otherwise known as Leonardo da Vinci, you can catch a direct train to Termini station. Trains run from 06.30-11.30. The journey takes 30 minutes and costs 15€.
Alternatively take the FM1 train lines. Trains run from 06.00-11.30. The journey takes about 40 minutes and costs 4€ one way.
Several buses including a night bus also take you into the centre. The journey time is around 50 minutes and costs approximately 4€.
If you’d much rather just jump in a taxi then the price will be around 40€.
There are also several buses from Rome Ciampino airport to the Ciampino train station. They operate from around 06.00-24.00 and are cheaper when booked in advance online. From here you can take the FR4 and FR6 to the Termini station.
A taxi from Ciampino airport costs approximately 30€ into the city centre.