We've done rome for 3-4 days with a toddler. By the fourth day everyone needs a day off..
There are cobblestones and uneven paths everywhere in rome - pushing a pram/stroller/buggy is hard going.
I've only done Rome, I haven't done any other italian cities. We did the same stuff in Rome twice (once without kids two years ago, this time with three kids and grandma) - and enjoyed repeating our adventures.
I can recommend the Rome B&B association for a private apartment rental within the city - check the exact locations. We stayed near the colosseum both times.
http://www.b-b.rm.it/
The advantage of a private rental is you can eat/drink and be merry and if the child wants to run around they can...more room than in a hotel or motel...
We can also recommend in some ways the hop-on-hop-off bus, however, be warned that they have an unreliable timetable and waiting on the side of a narrow busy street in the cold/shade/wind for 45 minutes for the bus is not fun.
I would not expect taxis to have appropriate baby seats, or bother using them...
I would not recommend taking the public trains with a baby/stroller - you are an absolute target for pickpockets and beggars.
I would probably recommend the public buses with a toddler, but not with a pram.
So, my recommendation -
Fly to Rome or take the train.
Take the express train from Rome airport to Rome main train station.
Get accommodation between the main train station and the Colosseum.
Plan to do the colosseum and roman forum one day.
Vatican and St Paul's Basilica one day.
Trevi fountain/spanish steps/etc third day.
Use the hop-on-hop off bus for the two days (vatican/trevi fountain etc).
Take it easy.
There is so much to see in Rome.
As they say...Rome was not built in a day.
oh, and watch out for pickpockets/beggars...
Final tip - the people who offer tours in English outside the colosseum, may actually be legitimate tour guides- ask them for their ID - we did an excellent walking tour of Colosseum and the Roman Forum - you jump the queue, avoid the traffic, and you are free to leave the tour whenever you've had enough. They went slow enough for our children, and we managed it with a pram...it was I think 20 Euro for adults including entry and the toddler was free...