Help with Family trip to Italy in June

Is this a first trip to these areas?
And have you been to Europe before?
Reason I ask is that a common mistake for less experienced travelers is to try to do too much. Getting places will take longer than you think.
You want to spend your time seeing and doing what you are going there to see and do, not traveling between points A and B.

My only thought is that you don't seem to be considering travel time. Taking a train to Naples does not get you to Praiano. Nothing on the Amalfi coast is that fast. Then you are going back to Sorrento and then renting a car. That will take awhile too and then you plan to drive for about 4 hours. And at least when I was there about the same time, buses were often full and you then had to wait for the next one.

Personally, I think you are spending too much time traveling. I would eliminate Puglia and redistribute those days. You could add a day to Rome and Venice as well as to Amalfi Coast. (it isn't clear to me if you would have three or four nights to redistribute as you list days as in two places.)

I think your second itinerary flows much better, but I do question you're stay in Praiano, since
Your only transportation option is the bus. But I don't know what you wish to see on the Amalfi Coast. And Sorrento might be a better choice if you want to do Pompeii, Capri or Naples. I do think your nites in Puglia make sense as it on the way to Pescara and allows a glimpse of this fascinating area. Though eleminate it and adding those nights, 1 to AC and 1 to Rome leaving the third fir your drive to Pescara might be a better choice for your family.

The way you have formatted your itinerary--listing your travels days as time in both cities--is potentially confusing to others and may cause you to feel you have a lot more time at each destination than will materialize. For instance, in the second itinerary you indicate you will be in Puglia June 15-18, but you actually will have just two full days there. You'll arrive from Sorrento most likely in the afternoon on June 15 and spend some time settling into your new lodgings, so there will be less than half a day left for sightseeing. Then on June 18 you'll be leaving for Pescara, probably meaning no time at all for seeing any of Puglia that day. That would make for a very rushed trip to a fascinating area which could also be quite hot in June.

I would modify itinerary #1 so all your days in Rome are at the end

This is what I recommend on your previous post.

Day 1. 8th. Travel.
Day. 2. 9th. Arrive Venice. Probably jet lagged.
Day. 3. 10th. Venice.
Day 4. 11th. Venice.
Day 5. 12th. Venice-Naples. 10:25-15.34. 5 hrs.
Circumvesuvian to Sorrento 1:06.
Day 6. 13th. Sorrento/Amalfi.
Day 7. 14th. Sorrento/Amalfi.
Day 8. 15th. Sorrento/Amalfi.
Day 9. 16th. Sorrento-Pescara. Rent car in Sorrento, 3:30 drive. You might need a car in Pescara so it makes sense to rent here.
Day 10. 17th. Pescara.
Day 11. 18th. Pescara.
Day. 12. 19th. Pescara.
Day. 13. 20th. Pescara.
Day 14. 21st. Pescara-Rome there's a regional Train at 9:23-12:45 takes 3 :22. # 3229.
Day. 15. 22nd. Rome.
Day 16. 23rd. Rome.
Day 17. 24th. Travel home.

For your travel Thru Puglia, this is what you could do.

Day 9. 16th. Sorrento-Matera. About 31/2 hrs.

Day. 10. 17th. Matera-Polignano a Mare. 1 1/2 hrs

Day 11. 18th. Polignano-Trani 1H15. Visit, lunch then continue Trani-Pescara 21/2-3 hrs.

This gives you...

3 Nites Venice

4 Nites Sorrento. Extra night here due to 5 hour travel time.

2 nites Puglia

3 Nights Pescara

3 nights Rome.