You need to back up and take fresh look at your overall plan. Start by being honest with yourself about how many full, usable, non-travel days you have in Italy not including your arrival or departure days.
What are your arrival and departure days?
How far are you coming - flying in long distance on a redeye, or just coming in from someplace close/same timezone?
If, as you say, you will have 8 full days in Italy, most people would recommend that you have enough time for just two locations, maybe three, but that's stretching it. You will have 3 young kids along, which is going to slow you down significantly and limit how much you can do and how hard you can push.
You say you're flying in to Milan - that would be a fourth location. This all looks difficult - and that's without your hoped-for "day trip to Pompei, Positano and the Amalfi coast" (which I think is impossible).
Sorry, I know this is not what you want to hear. But you ned to be realistic about how much time you have and what can be reasonably accomplished in that limited time.
Edited to add: I now see that you are coming in by train, and Milan is not part of your plans (although you originally stated that you were arriving in Milan).
So you begin in Rome (arrive by train), and you depart from Venice. In August (ouch). OK.
I would take issue with your approach saying that you have 8 days there, and because everything is no more than 2-3 hours away, you just lump all train travel together and subtract a day, leaving you 7 full days to use. Unfortunately, you are not going to spend 24 straight hours on a train and then have zero travel time on your other days (it's an interesting way to look at it, but I don't think it's going to work out that way). Look at it this way...
I assume you will arrive in Rome sometime other than early morning, on "Day Zero" - that is, not counted as a full, usable day. So then you have:
- Day 1: Rome
- Day 2: Rome
- Day 3: Rome
- Day 4: Florence/Pisa
- Day 5: Florence/Pisa
- Day 6: Venice
- Day 7: Venice
- Day 8: Fly out of Venice
There are problems with this. First, for each location change, you need to account for the time it takes to get up, deal with basic needs, check out of your hotel, find your way to the station, travel to the next stop, find your way to your hotel, check in, get settled, etc. That will consume half a day right there - and that's assuming you are all (including your young kids) on your "A game", efficient and quick at everything, never get lost, no need for a break, etc. Maybe that's how your family rolls, maybe not, but that's the best case: you burn half a day each time you move, possibly more. So for every stop listed above, subtract half a day. That means you have 2.5 days in Rome, 1.5 days in "Florence/Pisa", 1.5 days in Venice. See the problem?
Just because someone is trying to sell you a one-day tour that includes Pompei, Positano and the Amalfi Coast doesn't mean it's reasonable to try and visit all three one the same day (with kids, in August). People will try to sell you all kinds of junk you don't want.
Honestly, given your constraints, your family along, and the heat of August, here's what I would do:
Rome
Florence
Venice
That's still spreading yourselves thin, but at least with the above, you'll live through it and might even have some good memories of your trip.