The main bus transportation is ATAC, which travels all through Rome. The bus stops are quite frequent and if you navigate in this web page, you can find out which bus to take to reach the destination you want from where you are.
Termini is the main Train Station in Rome, from which all kinds of trains depart and go everywhere in Italy and also out of Italy. The best, and most comfortable line is EUROSTAR, which is very punctual, fast and stops only on the main cities in Italy: Rome, Firenze, Milan, Venezia, etc.
There is also the other alternative which is the Inter City, which is a bit slower than the EUROSTAR, but faster than all the other train lines. It is also well connected and costs a little bit less. If you need more information on the trains, where they go and at what time, you can navigate in its web page shown below.