Also, I always try and plan around events. With some dates we could look at what is going on in each city and that could have some impact on the decisions.
This isn’t the trip you asked about, but it’s the classic trip with two extra options.
1) Depart US
2) Arrive Prague. I always suggest starting in Prague because it is the most crowded and exhausting in my opinion, so hit it while you are fresth.
5) Morning Bean or CK Shuttle from Prague to Cesky Krumlov. This breaks up an otherwise sort of long train ride and lets you see the fairey tale town of Cesky Krumlov / river / castle, etc.
6) Mid afternoon Bean or CK Shuttle from Cesky Krumlov to Vienna.
9) Mid day train, Vienna to Budapest
Now, if you think that’s too much time in Budapest (I don't) then I suggest that you take and early afternoon train out of Vienna to Gyor (1.5 hours). There you check into the Hotel Klastrom a converted old monastery. That evening you enjoy the historic downtown district, the next morning you take a taxi to the Archabbey at Pannonhalma. After about 3 hours a taxi back to Gyor, pick up your luggage and go to the train station for a 1.5 hour trip to Budapst.
Alternatively there are some good day trips and overnight trips out of Budapest and Vienna. Melk for instance is a good use of one day from Vienna; and would justify an additional day in Vienna.
This is just my preference based on what interests me. For me, Prague is too compact and so full of tourists that I don’t think I am experiencing anything Czech. Vienna for me is a little too overly restored and as a result seems sort of plastic. Budapest is my love and I return several times a year. So, yes, I am very, very biased based on what I enjoy.