There are two tips I have to give you:
The first one is :
WATCH YOUR WALLETS EVERYWHERE! Rome is full of pickpocketers, scammers, especially in high touristy areas, and purse snatchers.
So be very careful.
On buses and subway and in crowded areas, keep your hands in your pockets, where your wallet is, and keep your bags and cameras across your shoulders and in front of you, as they have purse snatchers on scooters too. If you get crowded by somebody, in the bus or in front of a monument, beware that they might be trying to steal your wallet.
If you see gypsies coming towards you, go to the other side of the street. Don't let them get near you, especially the kids, they are trained to steal.
This is a very serious issue, and they can spot a tourist a mile away. so always be aware of your surroundings.
Don't wear jewelry or flashy watches, no large cameras, no wads of cash on you. As far as cash, you'll be better off using credit cards and deal with bills when you get home. On that note, inform the card company and bank that you will be abroad, so they don't lock up your cards thinking they were stolen.
Do not leave anything unattended ever, it will be taken.
Do not hang your bags off the back of a chair at a restaurant or any place in public. Keep them on you all the time.
If you carry anything valuable in a bag or backpack, keep it in front of you.
Keep the camera on a strap across your shoulders and in front of you. Around your neck is too easy to pull off. And back up your photos often, or change the memory cards often, so if one is stolen with the camera, you don't lose all the vacation photos... For backing up, hotels have computers and there are internet cafes where you can back up your photos in google accounts or other photo sites accounts. You can even create a gmail account just for the photos, and upload the photos there, so that your personal info won't be exposed to public places with your personal emails, but you can save your photos daily.
Your phone: make sure you keep it in a safe place and if you take it out hold it in a way that it cannot be easily pulled off your hand.
It's better to be extra careful and aware than have to spend time in a police station reporting the theft of your valuables and the loss of your money and passport. Leaving things in the hotel, unless you have a safe, is not much safer. Have a copy of your passport, licenses and cards in case the originals are stolen, so that you'll have some form of ID and it will make it easier to replace.
Try to stay away from solitary places at night, parks, deserted areas etc. If you have lived in NYC for any length of time, this should be a given, but people tend to romanticize Italy and are blind to the dangers.
So, use a money belt, if you have money to carry, or a neck pouch under your clothes, for most of your valuables. And carry with you small amounts of cash in your wallet, kept in a pocket where you can keep your hand on it. If you need something out of the neck pouch or money belt, go to a bathroom in a bar or restaurant and get it out there. Never ever take it out in public.
If anybody offers to help with tickets or luggage or anything else, tell them no politely and do not let them help you with anything. At best they will expect a tip and at worst they will try to rip you off.
We had that happen to us last time we went to italy. This man was trying to sell us trinkets and tried to engage us in conversation talking about where he was from and where we were from and tried to give us one of his trinket as a gift. We refused and he was somewhat offended, but we told him we were not interested in his "Free" gift and he eventually walked away. Never ever take anything from anybody, no matter what they say. If they shake your hand or engage you in conversation or put something in your hand, they will expect you to give them money for it.
Here are a couple of sites that share more info and some solutions. Check them out.
https://community.ricksteves.com/travel-forum/italy/pickpocketing-and-gypsies (A discussion and some pointers)
http://www.reidsitaly.com/planning/safety/gypsies.html (An explanation of the potential danger)
https://www.ricksteves.com/travel-tips/theft-scams (Ways people will try to scam you)
Well, you get the idea. Be careful, BE CAREFUL, BE CAREFUL!
If you are, you will be able to enjoy the trip without much loss.
The second tip will be in another page, called FOOD.