25-day trip in July-August

Most people overlook all the sights in Vienna. There is a ton to see and do (but it depends whether those interest you). I start with a fairly exhaustive search for sights along my planned route. You can use guidebooks, novels, history books, tripadvisor sights, recommendations from friends to build a list then decide which interest you most. The sights I'm most interested in inform both my route and schedule. I always end up paring back to only things that top my list (and wish I had more time).

Typically, I plan travel days (going to bed somewhere different than you woke up) and tour days (get up and go to bed in the same place). On tour days, I plan one big sight in the morning, lunch, then another big sight. I keep an additional list of sights (not must sees but "nice to see if I have time"). If I have time and energy after seeing my planned sights, I pick one or more of those with an eye toward not wearing everyone out.

On travel days, I try not to plan more than four hours of transportation (whether car, train or plane) because I've found that ends up being a full day (when you add packing, checking out, getting meals, finding your way, checking in, unpacking). I typically plan no sites or a mid-day stop at a sight where I'll also get lunch. Ultimately it's your vacation so go at a pace that works for everyone and see or skip sights based on your interests.

Take a look at leasing a car. It's only available through Peugeot and Renault but you can take delivery elsewhere. I picked up in Amsterdam and dropped in Rome. At the time the pick up and drop charges were much less than with rentals. Pick up on the way out of your first city and drop when you arrive in your last. Parking is a waste of money if you won't need the car to get around.

I traveled with five: self, spouse, 16 and 12 year old boys and an 8 year old girl. We were happy with a seven passenger car. We each had one carry on (girl had only a day pack). If we had been bigger or had more luggage we would have wanted more. I've found rental rates are good for small economy cars but the price goes up quickly for anything bigger or better. Leasing seems to make sense when you want a bigger and/or nicer car (I think related to resale value).

The best reason to have a car is to travel on your own schedule (particularly short stops along the way) and visiting sights that are difficult to reach by public transport. A car is cheaper than train tickets but can be an expensive pain in cities (for us Venice and Florence especially).

A lease price includes full CDW insurance, good if you had planned on it anyway. I've been using an Amex card insurance option that is a better deal (but I'm also renting tiny economy cars now).

Do you really need the car every day? You can't drive in Venice. Prague, Vienna and Budapest are also easy walking cities and require multiple days. They also have inexpensive trams and buses. Some train lines offer discounts with group purchases of 4 (and 6 year old would be a child rate). I would do some research on train costs vs car (remembering to add in cost of gas and parking).
I did a quick trip on rome2rio for this circuit and buses (which seem to be the same time or faster than trains) may be as low as $157 per adult.....but that is just a ballpark and may be off depending on whether seasonally priced or not. Even if $200 each, with youngest at half the cost, can you rent a car all told for less?
You also need to see whether a rental car allows entry free to all your countries of interest.

If your intent is to hit the cities for only 2 or 3 days a piece, and do road trips between, then a car is logical.

While 25 days is a generous vacation, you have six major destinations planned and one smaller one. You can spend at least one day less in Zagreb than in the other majors, maybe even two. I liked the city a lot, but a full day will allow you to walk most of the historic area. After that it's mostly museums, which you've indicated are not of interest to you.

I trust you realize that the car will be of no use to you in Munich, Prague, Vienna, Budapest, Zagreb and Venice; you'll just be paying to park it. Given your lack of interest in museums, I'd suggest you give serious consideration to changing the focus of the trip from major cities to smaller spots. With the except of Karlovy Vary, about which I know nothing, your present itinerary is pretty easily doable by public transportation. (Given that you're going to need a good-sized car, I think trains might even be cheaper, but I am not knowledgeable about European rental rates.) Why not make fuller use of the car by seeing some smaller places that might be less efficiently linked by train than the major cities currently planned. Side benefit: You wouldn't spend so much time fighting big-city traffic on the way to and from your lodgings. Second possible side-benefit: You might save money on hotels.

It may be very hot during you trip. Be sure all your lodgings are air conditioned.

Thank you all for a thoughtful replies. I was looking into parking a car on a long-term parking lots outside the city (under $2 a day in Budapest from what I've found). I have to see what options are available in other cities, including Venice, but with five people I feel that the car will be cheaper and would allow us to go to more places, national parks, etc.
I am also concerned about Zagreb-Venice leg of our trip. I'm not committed to going to Zagreb, I really need a nice two stops between Budapest and Venice, maybe mountains in Austria would be even better, I thing that we'd have enough of small European cities, squares and castles by then.
I would appreciate any ideas on Budapest to Venice route over two or three days.

A car large enough for five people and five people's luggage isn't going to be a small, cheap rental. Train fares in that part of Europe are generally very reasonable, so I'm not sure the car will be cheaper even before you consider the $$$ cost of fuel. When you price out car rentals, be sure you're including whatever insurance coverage you will need for peace of mind as well as the cost of supplemental drivers if you plan to share the load.

I've just taken a look at ViaMichelin's proposed driving route from Budapest to Venice, and it goes right past Ljubjlana, which is a wonderful city. It's much smaller than Budapest with a very walkable historic center stretching along both sides of a narrow river. It will definitely have lots of tourists, but it's a charming place. There's a castle right in town.

I am more concerned with convenience than $, so unless it's prohibitively expensive or extremely inconvenient, we're going with the car. Ljubljana is great, but we need one other stop in between Budapest and Venice.