From Vienna to Prague

I would give at least 4 days to Vienna (where I've spent 5-7 days in previous visits) and 3 to Prague. In between, Cesky Krumlov is a popular choice, which you can see will with just a one-night stay. It's a detour from the main rail line, but also served by bus and private van shuttles. Have you picked up Rick's guidebooks for Prague and Vienna? Each covers several side trip opportunities from the big cities. Austria's Danube Valley is also a logical route and nice for driving.

How many days would you spend in each city?

If you are interested in "Old Europe" I would split the time evenly. If you are interested in history, architecture, culture and the arts I would spend more time in Vienna, simply because the city was more important (Vienna ruled over Prague for a long time) and traditions are much better preserved.

Is there anywhere you would advise us to stop and explore on our way
from Vienna to Prague for a not to be missed experience?

There are many interesting places between the two cities, but most of them are difficult to reach by public transport, AFAIK. There is the Danube valley, the so called Wachau, of course, but this can be done as a day trip from Vienna. A town I really love is Telc, a cute little place with a fantastic town square. Mikulov is another nice town, and the Lednice–Valtice Cultural Landscape is a World Heritage Site.

Both cities have a lot to offer, including day trips. I would lean toward a little more time in Vienna to get over the jetlag.

I am currently in Bratislava, so I am going to pitch for a night here. The city centre is pleasantly devoid of tourists and full of bright young Slovaks eager to chat. Bratislava has the young city in a young country vibe that escapes both Vienna and Prague.

Also, I wholly recommend doing a "Post Socialist Urban Tour" with Authentic Slovakia tours. They pick you up in an ancient Skoda car and then bring you to some of the incredibly ugly Communist-era neighbourhoods, going into lots of 20th century history on the way. Very interesting tour, and a nice team of guys running it, so it is worth considering for something different.

Also, it is a most to spend at least a night in my beloved Brno and go to see the incredible Villa Tugendhat (needs to be booked in advance) and the awesome Moravian Karst underground cave network.

Vienna - Bratislava - Brno - Prague. That is how I would do your trip.