Here are some practical suggestions about how getting to the city centre from the airports and form the most important railway stations:
From Rome Fiumicino airport:
you have many choices, but the most common one is Leonardo Express, a non-stop service connecting this airport to Termini station (departures every 15 minutes and 32 minutes travel time); or you can chose among different bus companies and lines, according to your zone/address of destination. In case you prefer a private transportation, taxi have fixed rates according to the area you need to get to, or pick up the NCC service (car rental with chauffeur).
For more details visit the official website: http://adr.it/web/aeroporti-di-roma-en-/pax-fco-to-and-from#
From Rome Ciampino airport:
different bus companies and lines connect this airport to different districts of Rome; a special bus line offers the connection to Ciampino railway station, from where you can easily get to Stazione Termini. As alternative, also form Ciampino aiport you have taxis with fixed rate and NCC service.
For more details visit the official website: http://www.adr.it/web/aeroporti-di-roma-en-/pax-cia-to-from
From Stazione Tiburtina:
Tiburtina Station has good connections to trains arriving from the major Italian cities, and from Fiumicino airport as well; also some local trains serve this station (among the most important in Rome), and in front of this station is the most important bus station for outgoing destinations.
To reach the city centre, you can easily take the subway Line B (check the map to find ut your destination). Of course it offers both taxi and NCC services.
For more details visit the official website: https://www.stazioneromatiburtina.it/en/
From Stazione Termini
Termini Station is “the heart” of Rome public transportation: both national and local trains arrive and depart from here, it is the only subway stop connecting Line A and Line B, in front of this station (Piazza dei Cinquecento) is the most important bus terminus of the city: from here, you can (almost) go everywhere. And if this is not enough, you also have taxi and NCC services.
For more details visit the official website: https://www.romatermini.com/en/
Last suggestion: if you are not going to central Rome, then a car for rent may be the best solution, as some destinations could be difficult to reach by public/shared transportation.