Safety tips for solo traveler

This may be covered in the guide you read:

Apply the same level of caution like you would traveling solo to unfamiliar places in the United States
Don't stay out late to the point where there are no people on the streets.
Take care when crossing the streets.
Carry passport extra cash in pouch under your shirt.
Try not to fall asleep on the trains.
Avoid streets solicitations and ignore children asking for money.
Do not acknowledge complete strangers that ask "Do You Speak English"

In terms of safety, Italy is very safe and violent crime is very rare and decreasing.
The only risk is petty theft and pickpockets.
Your hotel safe in your hotel room is certainly safer than your back pockets or your purse. So take out with you only what you need for the day.
Keep credit card tel numbers separately in case you need to cancel them if lost or stolen.
The most common place for pickpockets to operate is aboard crowded public transport (buses, subway). Most of the time you will be walking, but if you need to take public transport be careful then. Taxis are obviously safer than buses, nobody will pick your pockets inside a taxi.
High speed Intercity trains are safe too. They aren't crowded and they are to expensive for pickpockets to travel on.
Be careful at any crowded event however (concerts, museums, etc.)
Be aware of people begging for money, offering help, etc. sometimes it's a gimmick for getting you distracted while an accomplice takes your handbag or inserts his hand inside your purse.
Carry a whistle with you (like in a key chain). If you need to call attention and need help a whistle is very effective.
Avoid the area near the Termini station in Rome at night. It has become a casbah full of young male migrants annoying solo women with their unwanted attention. My female friends and relatives in Rome avoid going there at night unless necessary or accompanied by a man, you should do the same. If you do go there, avoid dark empty streets. Stick to the main streets with lots of people.

Most of the places you mention have lots of people out and about until late. If you don't want to walk or take a bus/metro back to your hotel, ask the restaurant or bar staff to call a taxi for you.

My top suggestions for your trip.

—Treat each city the same way you would treat any major US city.
—As others have said, your biggest challenge is pickpocketing in tourist areas. The very same thing can happen at crowded tourist spots in the US. So again, think how you’d travel solo in San Francisco or NYC or Chicago, although you are probably safer alone in Italy than in many major US cities.
—Your hotel safe is your friend. If you do not need to carry something for the day - don’t. Lock it in your room.
—Print a spare color copy of your passport and keep it separate from your actual passport. (The reason I say print it - if the copy’s on your phone, and both your phone and your passport are in the same place, you may lose both.)
—Do not carry all your credit cards in one place. Same thing for cash.
—Having said that, keep a small amount of cash readily available. If you lose that, you still have your backup cards and cash.
—Do not leave your bag with your cards/cash/passport/etc., slung over the back of your chair. You can’t see what happens there.
—You are most vulnerable when on public transport and surrounded by crowds. Keep your hand on the zipper of your bag at these times.

And at the end of the day, you’ll be fine if you’re aware of your surroundings and practice good travel sense.

Is this your first trip to Italy? First time traveling solo?

It's dark here by 7 pm now. I walk alone. I'm not in the company of a man. I don't have that luxury. Some of us don't. I walk around Termini a lot in the dark. I walk all over the city a lot in the dark. Sometimes people are around, other times I'm alone. And I'm OK.

The Pickpockets are still here, but I'm not as buttoned up as I used to be. I was here in the August heat and had to loosen up a lot. And I'm still Ok.