The best way to experience Central Europe is by train, which happens to be my favorite way of travel in Europe. If you have never experienced other than Amtrak when it comes to train travel, getting around Europe by train might seem like a challenge (it is not). But I will write about train travel in Europe some other time.
So you want to travel to Central Europe (Austria, Germany, Czech Republic, Hungary). I recommend trains. Modern, fast, luxurious (if you choose to pay a bit extra), comfortable at night (if you have to travel at night), trains will make even an unlikely trip such as Paris - Berlin - Prague - Budapest possible. While you might want to set aside 10 nights for this particular trip (2 in each city, 2 on the train), this particular vacation would allow you to cut across Europe, while visiting 4 unique cities, 2 of them (Budapest and Prague) having only recently unleashed their potential.
Of course, a perennial favorite is the Prague - Vienna - Budapest package. Three particularly musical cities, with enough music and history to keep you busy for months.
For those short on time, two-city combinations might be the way to go:
Berlin - Prague; Prague - Vienna; Prague - Budapest.
If you are a big fan of Biedermeier, Vienna has the perfect exhibition for you at Albertina, one of the most important museums in the world. Albertina "is also the largest Hapsburg palace in the centre of the former K.K. Monarchy’s (royal and imperial monarchy) capital Vienna."