“Days” can be confusing, so let me first see if I understand:
DAY 1 Depart the US
DAY 2 Arrive Budapest and get to accommodations at 2 pm. You have half a day complete with jet lag
DAY 3 Budapest
DAY 4 Budapest
DAY 5 Train to Vienna. Morning train will get you there by noon. You have half a day in Vienna
DAY 6 Vienna
DAY 7 Vienna
DAY 8 Train to Prague. You have half a day in Prague
DAY 9 Prague
DAY 10 Prague
DAY 11 Depart Prague
If my assumptions are correct it’s a pretty rushed trip. But let’s start with the first question. I would go as early in June as possible. It will be cooler and less crowded and there are a few more options for cultural activities if that interests you. Remember European air conditioning isn’t all that frequent or great when it does exist.
Next, a car will do you no good in any of these cities, but will cost you a lot to park. Take the trains. Easy and relatively cheap.
Next I would reverse the route and start in Prague and end in Budapest. Budapest has more of interest to kids within the tourist zone and it will give them a reward for being good in Prague and Vienna.
Two ideas to consider:
DAY 1 Depart the US
DAY 2 Arrive Prague
DAY 3 Prague
DAY 4 Prague
DAY 5 Morning Train to Vienna
DAY 6 Vienna
DAY 7 Morning Train to Budapest
DAY 8 Budapest
DAY 9 Budapest
DAY 10 Budapest
DAY 11 Depart to the US
Better I would skip Vienna as I think it might be the most boring for the kids and I would do:
DAY 1 Depart the US
DAY 2 Arrive Prague
DAY 3 Prague
DAY 4 Prague day trip
DAY 5 Prague
DAY 6 Morning Train to Budapest
DAY 7 Budapest
DAY 8 Budapest
DAY 9 Budapest day trip
DAY 10 Budapest
DAY 11 Depart to the US