Bus number 175 (direction Partigiani) leaves from the piazza in front of Termini station and takes you to Piazza Barberini. It's four stops, but if no one's waiting or getting off, you might miss one of them. Tell the driver you want to get off at Piazza Barberini.
This web site will help you find out how to get somewhere with all means of transportation:
infopoint.atac.roma.it/bw.asp?lingua=eng
You can choose the type of transportation you want, for instance, avoiding the metro. You can also choose whether you want the fastest way, the way with fewer changes of bus, or the way with less walking.
You need to put in the address of the place you're leaving from and where you want to go. If you don't know the exact address, put in the nearest piazza.
Make sure you put the time of day you'll be traveling.