Croatia for the first time

Zagreb has Croatia's major airport, but there are some flights into Split and Dubrovnik as well. Most likely also into places like Pula and Zadar. You can use Google Flights to explore potential flight paths from your home airport to each of those cities. One might be notably quicker and/or cheaper than the others. In 2015 I was able to fly home (to Washington-Dulles) from Zagreb with just one stop (in Europe). Fares depend a great deal on the origin airport as well, so if you run into horrendous number for Croatian cities, you might want to consider flying into Venice. (See below for comments on moving between the two countries.)

I like Zagreb a lot. It has a very nice bi-level historic district and many interesting museums (though I have read here that many of the large museums in the lower town are still closed due to earthquake damage). However, folks on short trips to Croatia tend to want to focus on the coast and an island or two.

How many nights do you plan to spend in Croatia?

What parts of Italy do you want to visit? You mentioned Bari, which is down in Puglia, a very interesting area that might especially appeal to you if you haven't been there before.

It's not as quick to travel between Italy and Croatia as between many other pairs of contiguous countries. You might begin exploring your options on the website Rome2Rio.com, but you need to give specific origin and destination points. It's critical to understand that the actual fares, travel times and frequencies displayed on Rome2Rio.com are very often (I'm tempted to say "usually") quite inaccurate, but the suggested routes normally do exist. What you need to do is keep clicking down through the website until you find the company operating each travel leg you are considering, then go to that company's website for accurate information. Rome2Rio early always give you a link.

There isn't rapid, direct rail service between Venice and Zagreb or the Istrian Peninsula, as one might hope. You can take Italian trains to Trieste, and then it gets a bit more complicated. Croatian rail service is somewhat limited. As you can see from the map in that article, there's not a lot of track in the parts of the country most tourists gravitate to. There are nice buses filling in the gaps.

In my limited experience the BusCroatia website seemed accurate, but as I've told other travelers to that area, when it comes to buses in the Balkans, the only schedule I (almost) totally trust is the one on the wall in the bus station. Always verify bus schedules once you get to the city from which you plan to depart.

Skyscanner.com is a good resource for exploring intra-European flight schedules. It appears that the following Italian cities have non-stop flights scheduled in September 2022 from at least one city in Croatia, though not necessarily every day: Rome, Venice, Naples, Milan, Bari, Bologna and Turin. The Croatian cities with links to one or more of those Italian cities are Zagreb, Dubrovnik, Split and Zadar. I assume there are not links between all the possible pairs. For any single origin or destination, you can check on service availability to the other country by specifying that entire country rather than running through its cities individually.

I haven't taken a ferry in ages, but two websites suggested by Google are Directferries.com and Croatiaferries.com.

Renting a car in Croatia and dropping it in Italy would probably be extremely costly. Whether a car would be advisable in either country depends on what areas you want to see. A car is certainly useful in Puglia, and a car can simplify getting to the wonderful Plitvice Lakes National Park in Croatia (though there is bus service from Zagreb and Split).

The company GoOpti runs vans between Slovenia and Italy. I don't know how much, if any, of Croatia it serves.

You can take a ferry from Dubrovnik to Bari, So you Fly into Zagreb and work your way south along the coast, this might be a good option for getting to Italy. That is if you aren't pressed for time. The ferry takes a bit over 10 hours and obviously can take longer due to weather conditions. Actually, as I research this I'm not sure if that route is being serviced anymore. It does seem that you can take a ferry from from Split to Ancona and it is an 11 hour journey serviced by the ferry company Jadrolinija.

Train service in Croatia isn't the best but their bus service is pretty comprehensive. I just spent two and a half weeks in Croatia the end of September and I used getbybus.com to find my bus tickets. The schedule on there was always accurate event if you couldn't buy the ticket online. It can sometimes say "online sales are closed" for a certain route and that usually just means you have to buy at the bus station.

As stated above, the main airport is in Zagreb, but Zadar, Split, and Dubrovnik all have airports as well that are better serviced by EU flights than transatlantic ones. Many people say to skip Zagreb, but I quite liked the city and the change of pace from the coast. Definitely worth staying two nights if you can afford it time wise and if it ends up being the cheapest/most convenient city to arrive in. I took a tram/bus combo from city center to the Zagreb airport but I believe it was about 50 euros for a taxi. Its a 4 hour bus to Zadar from Zagreb and it can easily be broken up by a stop over at Plitvice lakes on the way down. I really recommend Zadar as its much much less crowded than Split or Dubrovnik but still has a ton of charm and is a beautiful city,

This is a bit of advice you didn't ask for but I think is worth nothing about Croatia, if you pull money out of an ATM, use the green ones from OTB bank. There are a TON of ATMs in Croatia that are basically scams, charge insane fees, and give terrible exchange rates.

Driving into Croatia via Venice and Slovenia is easy (pre-COVID) and quick. We enjoyed Rovinj, Motovun, and Grožnjan as well as Ljubljana, Bled and Piran in Slovenia back in the Summer of 2017.

the best way to Italy for the remainder of the trip

That very much depends on where in Croatia and where in Italy. But if you destination is Bari it is probably a very good plan to end up in Dubrovnik and take the ferry over to Bari.