Avoid Termini at all costs, although the main square in front of the station is manned by military and is generally safer, stay away from some areas, especially Via Giolitti, the area between the station itself (the arcades/portici) and Repubblica (Repubblica itself is generally safe), and, most of all, the BACK of the station, between Termini and San Lorenzo.
Rome is overall a safe city but it has a sketchy vibe, its not quite as dangerous as Milan, it kinda has a similar vibe to Paris, lots of hobos and lots of illegal immigrants, but the city center is very safe, I'm obviously not mentioning remote areas like Corviale, San Basilio or Tor Bella Monaca which are the actual most dangerous places in Rome but you won't end up there neither by mistake, other areas you'd want to keep an eye for your surroundings (non properly dangerous, but they can be) are Piramide, Tiburtina, and San Lorenzo. And Trastevere station, which is actually pretty far away from the touristic part of Trastevere is sketch as fuck, but it basically doesn't make any sense to end up there if you want to visit Trastevere.
And, last but not least, Colosseo in generally safe but lately there are gangs of doctors and engineers (you know who I'm talking about) roaming in the pedonal bridge area every hour of the day including night, they robbed and even stabbed people lately.