Eastern Europe travel plan

I think you have the start of a great itinerary for your 23 days! To see the distances between cities via train check out the website Rome2Rio, type in each leg of your trip and it will show you travel times for a wide variety of transportation options between each, and approximate prices for each leg.
Your locations can be arranged in somewhat of a circle.
I would fly open jaw in to Berlin and out of Prague. With 23 days, you can certainly fit it all in if you don't linger more than two or three days in any one place.
I would travel Berlin to Warsaw, Warsaw to Krakow, Krakow to Budapest (I would fly this leg...10 hour train ride otherwise), Budapest to Vienna, Vienna to Salzburg, Salzburg to Prague (stopping off in Cesky Krumlov on the way for a night or two).
Get out a piece of paper and plan out each day of the trip including travel time and how many days you want to spend in each location. It should give you a good sense of the pace of your trip and if you need to modify anything before buying your tickets.

I will be a little more opinionated. A reasonable route might be
Berlin
Warsaw,
Krakow,
Budapest,
Vienna,
Salzburg,
Prague
That would mean 7 days of traveling and 16 days of sight seeing (not that the traveling wont have some sight seeing value). So seven cities in 16 days is going to be 3 days in two cities and 2 days in the rest. For some the experience of the trip is enough to make it worth while, for others they want to become more intimate with the destinations. Personally I would do:

1 UK to Berlin
2 Berlin
3 Berlin
4 Berlin to Prague
5 Prague
6 Prague
7 Prague to Cesky Krumlov
8 Cesky Krumlov to Vienna
9 Vienna
10 Vienna
11 Vienna / Melk
12Vienna to Gyor/Pannhalma
13 Gyor to Budapest
14Budapest
15 Budapest
16 Budapest
17 Budapest to Krakow
18 Krakow
19 Krakow
20 Krakow to Warsaw
21 Warsaw
22 Warsaw
23 Warsaw to UK

Too tight, my first cuts would be Melk then Gyor

Berlin and Warsaw, for instance, both have lots of cheap flight options to/from London. From Krakow, however, most flights include a connection, except an early morning RyanAir (nobody's favorite) flight to Stansted. Explore more at www.skyscanner.com.

Rick’s Train Travel Time & cost Map gives you an overview of faster train travel times in hours between big cities, as well as regular (full-price) 2nd-class fares. Buying tickets as you go in stations is not difficult. Several of these train routes don't need reservations, but to/from Warsaw and night trains from Budapest to Krakow would be reserved.

You could also consider a European East pass that starts at $195 for 5 days of travel in Czech Rep, Austria, Hungary, and Slovakia. It's more of a convenience than a cost savings. The only pass that covers more countries (Eurail Global) is too expensive to use for this route.

I want to thank everyone for their help...And I hope I meet you during my Summer trip!

Patrick

Depending on what day you are in Budapest, you just might run into us.

That would be awesome! Look for a tall guy who isn't a spring chicken!

Patrick