I hope you are not trying to squeeze all these destinations into a 15 days itinerary - do you mean you are staying in Frankfurt for 15 days and then visiting the other towns? These are not exactly day trips from Frankfurt.
In general, if visiting cities in central Europe (as opposed to rural areas), you do not need a car. It may be a cheaper choice for 4 people if the car is returned in the same country where you picked it up, but less comfortable than a train. Check train schedules on bahn.com - but I suggest re-considering your choice of destinations if you only have 15 days. There is a lot to see closer to Frankfurt than Budapest or Vienna...
That's a lot of major destinations for your time frame. I think dropping one would make it less of a whirlwind. Train will be quicker than driving, but it's still a lot of ground to cover. bahn.com gives you timetables
Do you fly home from Frankfurt too? That means spending the last night back in Frankfurt.
Obviously it depends on everyone's interests but I'm surprised that Berlin (which these days has a reputation as a "cool" destination) wasn't on the list, although Prague/Vienna/Budapest ("The imperial cities") is a common combination. In fact we did it for our honeymoon, on an organised tour (this was back in 1991 when tourism was less developed especially in Prague and Budapest) , starting and ending in Prague, and the trip lasting two weeks.
I suggest getting the family's input on this too.
How old are the teenagers (13 to 19 is quite a range), and what kind of holidays/vacations do you enjoy? This itinerary suggests an interest in history and visiting ancient buildings and museums. That is how we spent most of our holidays with our son through his teen years but I have to say that some friends used to be surprised that he found it enjoyable!Edited: 5:40 pm, June 06, 2018