Edited
The only bad thing about using strollers in Venice is the bridges and maybe getting on and off of vaporettos safely, however, on the waterfront in front of St. Mark's Square the bridges do have ramps. Just don't bring one of those huge bulky and heavy strollers if at all possible. Rome won't be such a problem, either. Yes, it's not a wheelchair/stroller friendly place for sure, and all the museums and churches will have steps to navigate, but it won't be impossible. Some museums, such as the Borghese, are strict and probably won't let you in with a stroller at all. I have found that there are no hard and fast rules in this regard. Every church and museum has their own rules. You may want to bring an "umbrella stroller" to use in those places where a bulky stroller might be prohibited. In Rome, during rush hour, a stroller will be a hindrance as the buses are EXTREMELY crowded. During these times, a taxi would be a better alternative.