Post Sicily trips to Asti and Bari

First You are flying north for 1,000 kms, from Sicily to either Turin or Milan. Then you are going to take a train headed to a city that's 800 kms south, one week later.

Did I got it right?

Yes! We’re staying for a month this time, and we’re trying to maximize our time there.

Last time we stayed for three weeks plus a couple days and started in Rome drove to Tuscany back to Rome, train to Pompei drove Almalfi coast drove back to Rome and boarded a Mediterranean cruise that went to all the major cities on the Tyrrhenian sea. We figured we would take it easy this time and avoid some of the driving. 😜

It would be more logical to first fly from Sicily to Bari, then:

• fly from Bari to Milan or Turin (better Turin but fewer flights) for Asti
• OR take the overnight train from Bari to Asti (direct).

Daytime train options between Asti and Bari are not great (9-10 hours with several changes)...

You could also fly from Sicily to Turin or Milan for Asti, then travel from Asti to Bari, but this will make you travel a longer total distance.

Please share what is already booked. Are you flying out of Bari? Knowing what is already set will save us all from transportation suggestions that are moot.
Making assumptions from what is shared, it seems you would fly from your departure point in Sicily to Milan, then train to Asti. (BUT, driving in the Piedmont is so easy, and all those private drivers could cost as much as having your own car.)
Then you would return to Milan and fly to Bari. Have you done your wine research already?

fly to Bari

I think They want to take the 9-hours train from Asti to Bari to see the Adriatic coast.

Backtracking for 800 kms does not seen the best way to "take it easy", but we don't know where they are going after Bari.

No flights, ground transportation or lodging is booked yet. We're planning on flying from Catania > Turin, then take a train to Asti (spend a week here). Then, take a train from Asti > Bari (spend a week here). We're open to driving if it doesn't make sense to hire or use public transit. My wife is doing the wine research and is a WSET level 2 (read serious wine drinker), but we're open to must see wineries in both Asti and Bari. Suggestions on transportation, places to stay and see are all open for discussion!!! Thanks in advance.

We're going from Bari back to Rome (probably by train) to fly back to Chicago.

I would look at going to Bari first if you can get flights home from Milan. I'd do anything to eliminate that much travel in the middle, but you clearly chose those two places for a reason, and both are lovely.

Ah, if your flight home is from Rome (and cannot be changed to a flight out of Milan), starting with Asti or starting with Bari makes little difference. Do whatever works best in terms of flights and trains!

The reason we decided to go north after visiting Sicily and come back south is because the costs of flying from the north were prohibitive (~$3,000 USD) versus flying to and from Rome (~$1,200 USD).