"Base" for such a trip, you can choose Rome. From Rome 1, 5 hours by train to Florence - the day for an inspection and in the evening to the "base."

To Venice from Rome is far away, it may be just the move that can be allowed with the child on such a trip. You can book a couple of nights in Venice and from there fly to Moscow. There are flights Moscow - Rome and Venice - Moscow. For example, Transaero offers an affordable price on this route.

Many tour operators offer a group tour for 7 nights, for example, Italy - Switzerland - Liechtenstein: Milan - Venice - Vaduz - Zurich - Bern - Turin. If the tourist has time, then before or after the group tour, add not-entered cities - Rome and Florence, bring the total number of nights to 10 or more.

If traveling in the summer, then instead of living in Rome, I recommend to stay on the coast in 20 - 50 km from Rome