Travel to Prague and Vienna

This part of Central Europe has a lot to offer. Vienna is a wonderful city which can easily keep you busy for a few days. A day trip to Melk (easily done with the train) with its famous monastery won't disappoint you. Salzburg brings you almost in Germany. A little bit touristy, but nice for a day. Others will undoubtedly have suggestions for more things to do in this part of Austria.
Over the border you'll find Nuremberg (Nürnberg) an interesting city and a perfect base for worthwhile day trips to Munich, Bamberg, Würzburg or Regensburg (all within a one hour radius). Nürnberg is only two hours from Frankfurt, which is probably the airport you need for your flight back home.
The maps on viamichelin.com can inspire you for other possibilities and the pan-European planner of the German railways http://www.bahn.de/p_en/view/index.shtml gives all the connections (and prices for the trains they operate in or from Germany).

The classic tour for a lot of reasons is Prague, Vienna and Budapest. Two weeks is just about correct for the trip as well.

(Saturday) Depart the US
1. Arrive Prague; and I think Prague is the best of the three to begin the trip.
2. See Prague
3. See Prague
4. Take morning Bean Shuttle or similar to Cesky Krumlov
5. Take afternoon Bean Shuttle or similar to Vienna
6. See Vienna
7. See Vienna
8. Take morning Train to Budapest
9. See Budapest
10. Day trip to Szentendre
11. See Budapest
12. See Budapest
13. (Saturday) flight back to Prague on Czech Airlines, non-stop for under $150

This shows you three cultures, large cities and some smaller towns. None of the transfers are more than 3 hours so you aren’t sitting in trains or busses or …….

The suggestions in the posts above are all great. You should also look at Görlitz, Dresden, and Berlin (in another direction from Prague). Which of these is most worthwhile? Everyone has a different opinion after seeing them, so you should do your own research and see what appeals to you.

While trying to connect places, be sure to investigate travel times before making final decisions. Start with trains, which you'll find on the Bahn (German Rail) website http://reiseauskunft.bahn.de/bin/query.exe/en, following Rick's tutorial: http://www.ricksteves.com/travel-tips/transportation/trains/online-schedules

If the trains times are long, you can look for flights using Skyscanner: http://www.skyscanner.com/

Of course you have many possibilities, but I agree with others that Prague-Vienna-Budapest is a good two-week itinerary. Cesky Krumlov and Melk are worthwhile stops along the way if you have time. But there are many places to see in Germany and/or Switzerland. Only you can make these choices. My caution would be to limit yourself to just a few cities for several days each, rather than rushing from place to place so you spend too much time in stations, trains, airports, or planes, and not enough in the places you came so far to see.

Train is probably the best way between Vienna and Budapest, but you might consider the hydrofoil alternative down the Danube. I haven't taken it, but I have taken a longer cruise there and much of the river is beautiful.

From my limited experience, I liked Budapest the best of these cities, but as Harold often points out they're as different as apples and oranges and bananas. James is a well-known Budapest fan and perhaps his influence has rubbed off on me. ;-)

Dick, I am a "Eastern" Europe fan as much as Budapest. I just give better advice on Budapest. "Eastern" Europe is on the cusp of change. See it now or never see it as it is today. Time is running out. Paris, Prague, Vienna and now even my beloved Budapest have attained a point where they will be what they are today for then next several decades; but Romania, Bulgaria, Slovakia, Montenegro Ukraine still have one foot solidly set in the mid 20th century (and sometimes the 19th century) and that's where I think people will learn the most about the forces that shape or have shaped our world. ............................Did i say that? Never mind, I'm full of $hye^. Go to Budapest: http://www.cnn.com/videos/tv/2015/06/06/bourdain-cooper-budapest-sot-ac.cnn

There are a lot of possibilities with Prague as the starting point. Prague-Vienna-Budapest is a good bet if you want to stick to large cities. A nice thing about Prague-Vienna-Budapest is the train connections are really easy. Those all all great places to, visit, but there is a lot more to these countries than the big cities. In Czech Republic we enjoyed Cesky Krumlov, Telc, and Olomouc very much on our 2011 trip. The same trip included Sopron, Hungary, another winner, along with Budapest and Prague. Of the three big cities mentioned, Budapest is my favorite to just be in - lack of the tourist hordes that you will find in Vienna and Prague. Or, they are just more spread out in Budapest. I could really entertain myself for hours just walking in Budapest - walking across the river on its grand bridges and taking the tram up and down the riverfront. Enjoying sunset views from Castle Hill (the sun is at your back looking back over the city center). Budapest offers excellent value, and we found the people to be especially nice. Not a lot of blockbuster sites, but lots of humbler ones and that can be liberating. Vienna is a really great art city, which is very appealing for us. The big downside of sticking to the big cities is that you would really be shortchanging the countries in which they are located. It's hard to plan a Germany trip that remotely does the country justice over the course of a two weeks itinerary. Austria you could do on its own pretty well in that span of time, and two week in Czech Republic alone would let you stretch out a bit and see more. As much as I love Budapest, beyond Eger and Sopron not that much calls out to me, based upon my research and experience. It really depends upon how much you like big cities, whether you think you'll ever go back to these countries, and how much you have traveled in the past. The more experienced I get as a traveler the less I am oriented toward goal-oriented tourism, so I tend to gravitate toward covering less ground but going deeper and visiting places more off the beaten path. Oh and most of all, it depends upon what kind of trip you want - it's your trip!

VS, "As much as I love Budapest, beyond Eger and Sopron not that much calls out to me," You overlooked Pecs. Between Pecs and Eger, I prefer Pecs, its a beautiful town with great Roman and Ottoman history. Then there is Gyor and the Archabbey. Still I encourage people to shop around. Budapest, Paris, London, Bucharest, Belgrade, Sofia, Warsaw, etc...... Don't get tied down too fast. You can always return. After our first trip to Budapest we explored a half dozen other countries before returning to Budapest and we continue to split our trips with 6 days in a new place followed by 12 days in Budapest.

First: you got only 13 days. That's not many. Second: do you want to see more cities or dig deeper in just one (maybe two) country?
Without knowing you I would suggest mix of cities and small towns which are logistically in one line: Prague, Cesky Krumlov, Melk, Vienna, Budapest. Go for overnight in Eger if you have time. It's not clear to me if those 13 days you got include Prague or not. If you like classical music then this is your opportunity:
http://www.prague.eu/en/event/7516/prague-spring-2016-71th-international-music-festival