A flight + train is possible with 3 bags, a stroller, and a 4 year old, because we did exactly that two years ago. It's not great, but it is possible. Here's my advice based on how we planned it:
1. Only bring as much luggage as one set of hands can manage. The other pair of hands will have the stroller and the kid. 3 large bags was challenging for me, but I made it work by traveling first class on the train (more luggage space, but it was tight), and the advantage of speaking Italian and being very familiar with the cities I was transiting and the trains that I was riding. I would have been happier with two bags vs three, though.
2. Stay in one place the whole two weeks. We do this anyway, and I'm passing this on to you - rent an apartment in your primary destination, fully unpack, and if you want to go somewhere else for a couple of days do it without moving all of your crap. It costs more, but you do not want to move your stuff more frequently than the book-ends of the trip.
3. If going in and out of Rome and staying elsewhere, time your flights for the high speed trains that go directly to FCO. No transfers with luggage, period.
If you want to disregard all of the above, then the other thing you could do is hire a driver between all of these points. But for a two week trip, I would very seriously consider an apartment rental in Rome or Florence and rely on trains (without luggage) to go places. I would still opt for apartment rentals over hotels because of the 4 year old, provided that you can get by without the services of a hotel. If not you'll sacrifice some space for assistance. If going the hotel route, make sure and map out locations - many of the chains don't have presence in areas where you want to stay, as the goal should be to primarily walk to your places of interest.