I'm currently in Rome, where you are considering travelling. The situation here is normal, given there are very few reported confirmed cases (4 so far, including 3 that have been declared 'recovered').

In general, people are more cautious with being near each other but mostly people are going about their business as usual. For example, if you go to a supermarket you will see, especially in larger chains, staff generally wearing gloves to handle items. This is probably mandated from management, and it is in response to people concerned about the possible spread of the virus. You also see the occasional customer wearing gloves when shopping. 

Please bear in mind that Rome is relatively far from the northern part of Italy, where currently an outbreak has been declared and a number of towns have been quarantined.

Having said all of the above, nobody can give you 100% assurance that anywhere is safe right now. Just look at cases recently reported in both the UK and the US. Such cases do not seem to have any clear links to people who came from any of the affected areas.

You should perform your own risk assessment. Are you travelling with people who are in a category deemed to be vulnerable (e.g. people with chronic/established health conditions such as cardiovascular disease, diabetes, etc. / elderly people) ? If so, even though the risk of infection is low in Rome, the impact of that risk could be very high and it should not be underestimated.

Also, this is a rapidly evolving situation. If you are travelling for leisure and, during your stay, the situation in central Italy changes, you may end up being stuck here for longer than you planned, as authorities and/or airlines update their operational guidance. Check what happened to the passengers of the Diamond Princess cruise, or international guests in Tenerife, who are currently under quarantine for 14 days at least.
In that case, if you can't afford to be away from your base location for longer than you had intended you should consider what the impact of a forced prolonged stay is for you and your travel companions.