Vienna -> Prague -> Berlin

With only 10 days to play with, I would just visit Berlin and Prague, as you are travelling peak season and will lose at least half a day in transit between each location. Berlin could easily fill a week, so I would just cover this and Prague.

3/4 nights in a place is only 2/3 days and not enough to cover each properly. You need to crack on and book accommodation asap.

It's a good thing to leave Switzerland off this trip - you have just barely enough time for this itinerary. The cities are pretty well-connected by train rides of about 4 hours, so they match up pretty well. You're right to do Vienna 3, Prague 3 and Berlin 4 - it would be an awesome trip.

I pull my list of sites to see from a city-specific guidebook because I like an in-depth look at what there is.

Your itinerary looks borderline reasonable. It seems like you got 3 to 4 days for each city which is not much. Either you see more and quickly or fewer cities and deeper. You can consider (as it was already suggested) just Prague and Berlin. But it will be doable with Vienna, too.

I did the Rick Steves tour which was very close to this. The current tour is very close to what we did. So, yes, I think that you could do this.

Pam

Like this?

12 depart US
13 Arrive Vienna early afternoon (jet lagged)
14 Vienna
15 Vienna
16 6 hour door to door train trip to Prague – arrive mid afternoon
17 Prague
18 Prague
19 6 hour door to door train trip to Berlin – arrive mid afternoon
20 Berlin
21 Berlin
22 Berlin
23 Depart to the states

Sure, works. In Vienna you missed Melk and the Wachau Valley (at a minimum). In the Czech Republic you missed Cesky Krumlov, Karlstejn Castle, Karlovy Vary, Kutna Hora, etc. In Germany, well where does one begin.

For the amount of time you have I would:
12 depart US
13 Arrive Vienna early afternoon (jet lagged)
14 Vienna
15 Vienna
16 Melk and the Wachau Valley or one of many other great day trips out of the city. Here are some ideas: http://www.bigboytravel.com/europe/austria/vienna/topdaytrips/
17 Shuttle to Cesky Krumlov
18 Shuttle to Prague
19 Prague
20 Prague
21 Day trip to …. Here are some ideas: http://www.myczechrepublic.com/prague/sightseeing/day_trips.html
22 Prague
23 Depart to the states

Germany as a dedicated trip next year.

OK!!! I LIED!!!

First, because I am very biased and second, because you couldn’t get me in Prague in the high season.

For the amount of time you have I would:

12 depart US
13 Arrive Vienna early afternoon (jet lagged)
14 Vienna
15 Vienna
16 Melk and the Wachau Valley or one of many other great day trips out of the city. Here are some ideas: http://www.bigboytravel.com/europe/austria/vienna/topdaytrips/
17 Train to Gyor, see the old town, spend the night at the converted cloister, Hotel Klastrom
18 Visit the Winery and Archabbey at Pannonhalma, then late train into Budapest.
19 Budapest http://www.cnn.com/videos/tv/2015/06/06/bourdain-cooper-budapest-sot-ac.cnn
20 Budapest
21 Day trip to Szentendre
22 Budapest
23 Depart to the states

"In the Czech Republic you missed Cesky Krumlov, Karlstejn Castle, Karlovy Vary, KUNTA Hora, etc."
I noticed that more people spell Kutna Hora this way: Kunta Hora. Of course in this case it could be just an unpleasant typo. I already advised few times against this spelling. Look at my second post here:
https://community.ricksteves.com/travel-forum/czech/best-day-trip-from-prague-e3467f8a-09a0-4563-abb9-3676978c8a0e

Ilja, let's call it a typo. Fixed. Thank you.

To answer your question about reading the individual country (or city) guidebooks, I would say yes, definitely -- you can probably borrow them from the public library to decide which places appeal to you most, then buy your own copy to mark, rip up and carry (library REALLY frowns on that!)

I very much appreciate all of the input, it is truly helpful.

I'm headed to the library tonight to obtain some materials on the three destinations to try and determine if I want to pick 2 of the 3 (or 4, because Budapest is an interesting option I hadn't thought about yet) or try and squeeze three in, since it sounds like these three cities are doable with the proposed itinerary but at a material cost of not seeing the full suite of experiences that I would be closer to achieving if simply traveling to two cities.

I also appreciate the encouragement to get crackin' since it is at the height of the tourist season, which I tried to avoid but unfortunately our scheduling didn't allow for a May or August trip.

My off the cusp reaction would be to shoot for Berlin and Prague but the flights are less appealing from a time (16->24+ hrs) and/or price perspective (+40-50%). Either way, I've got some reading to do! Thanks again for the help.

Another option to consider is Prague, Dresden, Berlin. Ideally flying into Prague and out of Berlin to spare any backtracking. For us 2 days in Dresden was enough - I can easily spend a full week in both Berlin and Prague.

I also prefer the city guides for a trip like this as they have more in depth coverage and maps.

I don't see anything wrong with Vienna (3) >> Prague (3) >> Berlin (4)