Hello Diana,
here are some thoughts from someone who lives in Central Europe and has travelled extensively there.
All of your destinations are Tourist Disneyland places. In Prague, Český Krumlov, Budapest and especially in Kraków you will see exactly the same shallow collection of Best Irish Pubs, Traditional Restaurants and cheesy pavement artists.
As for transport: for the price of having a private driver you could be going first class on the excellent rail services between Poland, Czechia and Hungary and still save money. You would also pass through magical scenery that you would miss if you go by car.
I don't know how fixed your plans are, but here is a route I would suggest that would help you experience a more authentic Central Europe in 14 days without compromising on great destinations.
Tarnów 2 nights
Have a look at Tarnów on Google. It has one of the most exquisite city centres I've ever seen. Everything is walking distance. Spend two nights there recuperating from your flight. Enjoy the renaissance architecture and discover its Jewish heritage. This is the small town to replace Český Krumlov on your itinerary.
1st day: arrive Kraków Airport and get yourself to Tarnów, which is 50 miles east of Kraków.
2nd day: Tarnów. Sleep in Tarnów, then rise early to travel to Kraków.
Kraków 2 nights but 3 days thanks to overnight train to Prague
Kraków is lovely. Non-touristy but interesting places would be the Tadeusz Kościuszko burial mound: the Pole Kościuszko was a hero of the American War of Independence. Another interesting thing to see is the district of Nowa Huta, a communist model city gifted to Poland by the USSR in the 1950s. It is one of two socialist model cities in the world.
3rd day: Rise early in Tarnów, get train to Kraków. See market place, Wawel etc.
4th day: Kraków. Day trip to Auschwitz. Evening: Kazimierz.
5th day: Kraków. Day trip to Wieliczka salt mines. In the evening, overnight train to Prague. You can book a compartment for yourself.
Prague 3 nights
Prague speaks for itself. There is more to do here than you can manage in two weeks, but Vyšehrad is lovely, and all the Art Nouveau stuff. I'm sure you already know what you want to see.
6th day: arrive Prague early morning.
7th day: Prague
8th day: Prague. Sleep in Prague and in the morning get train to Brno. Is 2 1/2 hour train journey.
Brno: 2 nights
Brno is Czechia's second city and very beautiful, but much less touristy. It is the capital of Moravia and famous for its wine. Also you are well on your way to Budapest here, with direct trains to Budapest passing through. After Prague this will be more restful.
8th day: Brno. See the cathedral
9th day: Brno. Sleep in Brno and rise early to get train to Budapest. Is four-hour train journey.
Budapest: 4 nights
10th day: Budapest
11th day: Budapest
12th day: Budapest
13th day: Budapest
14th day: fly.
I hope this helps you. I do hope you have a great time, and do check back if you've got any questions.