You should do everything you would normally do in a large metropolitan city in order to ensure your safety.
Petty crime, especially pick-pocketing, is common in Rome. The best way to protect yourself against this is to only carry with you what is strictly necessary. Never carry large amounts of cash and avoid camera bags and expensive jewelry when possible. Pickpockets use clever ploys and often work as teams, in which one distracts you, either by being very friendly/helpful or by throwing/spilling something on your clothes, while the other picks your pocket or bag.
The squares outside the Rome Center (Campo de’ Fiori & Piazza del Biscione) and in Trastevere are gathering spots for large numbers of young people, especially in the summer. Occasionally there have been episodes of drinking related violence. You should be aware that this can take place and safeguard against possible problems by not walking alone at night and by avoiding contact with those congregating in the piazzas.