If you take a stroller, take a very sturdy one that can handle cobblestones and curbs, and it should be rather compact, as you’re going to be moving around in big crowds in narrow streets. You may want to carry her on your back in a backcarrier frame for events like the Mass at St. Peter’s Basillica.
Wear very, very comfortable shoes. Take two pairs so that you can alternate them, unless you literally have perfect shoes that will never rub or bother you at all.
Although the weather will probably be too cool for this to be an issue, if you ever want to go sleeveless, carry some sort of lightweight shawl to throw over your shoulders so you can go into churches.
Go into various churches on a whim. You can’t tell from the outside what you’ll find on the inside.
Don’t be afraid to be rude if a begger gets too persistant. It’s a common tactic to distract you by touching you or maybe getting between you and your child, etc. If a begger gets so far as to grab ahold of your arm (as happened to me), be very assertive and get away from them. There are absolutely beggers who will go into churches, climb into the pews, and grab your arms begging while trying to steal your purse. This nearly happened to me and the sacristan had to interfere violently. This happened at St. Susanna, the “American Parish” and I think they target Americans, who are naive and too polite to actually get physical.