Celebrity Edge Rome to Rome

aglavac, if you haven't done so already, you should probably sign up and post your questions on Cruisecritic as you'll be amongst more cruisers there than you will be here.

https://www.cruisecritic.com
https://boards.cruisecritic.com/forum/464-italy-ports/
https://boards.cruisecritic.com/forum/16-celebrity-cruises/
https://www.cruisecritic.com/rollcalls/?cruise_line_id=9&ship_id=1130&sailing_date=202010

Where else does the cruise stop?

Try toursbylocals.com. They have a section just for cruisers. We have used them while on a cruise in the british isles.

In Rome a nice day trip is Ostia Antica. Easy to get to by public transit and will only take about three quarters of a day. With regard to ship excursions, they are more expensive than doing things on your own. But, if you take one and are late getting back to the ship, it will wait for you. Not so if you are off on your own. And the traffic in some of the ports you are likely to visit have very bad traffic problems. Naples can be horrible at times. We've seen folks literally running along the pier as the ship was departing. Our rule of thumb is to do things on our own if we will be close enough to the port to hoof it back if need be. But for traveling good distances from the port we'll take a ship tour.

If you don't want to use the ship excursions, then you need to do your homework. We have done seven European cruises and rarely take an excursion but our homework is extensive. We research each port stop to id what we want to see and how to see it. We review guidebooks and travel DVDs from the local library. Sometimes a hop on/off bus/train is appropriate. Maybe hire a taxi driver for a half day. And generally a trip to the TI will find local walking tours. I could recommend a number of tours but without knowing your interests, I could be wasting my time. What do you want to see?