Prague-Budapest

You need at least three days for each city. You probably would like to stay in the center in both cities. There are plenty of hotels and apartments to choose from. Prague to Budapest is 7 hours by train. I haven't use a night train for this distance. You can break that long journey by train for overnight stay in Brno or Bratislava. Check these websites:
http://www.pragueexperience.com
http://www.praguepost.com

With five days I'd just pick one city.

Thanks for he help!

Thank you! Very helpful and really appreciate the web sites!

Given how long it takes to get from Prague to Budapest, I agree that with only 5 or 6 days you should pick just one, and planning on day trips or overnight trips from there. From Prague, you can daytrip to Kutna Hora or various castles, or spend an overnight in Cesky Krumlov or Olomouc. From Budapest, you can daytrip to Vac or Szentendre, or spend an overnight in Eger or Pecs. Except for Vac, details of all of these places (and how to get there) are in Rick's books.

The problem is that Budapest and Prague are VERY different (and Vienna, often mentioned in the same breath, is different from both of them). I love both Prague and Budapest (and am cooler toward Vienna), but many strongly prefer one over the others. Since you don't know how you'll feel about each one until you see it, you'll just have to read as much as you can and see which one appeals more to you right now. Failing that, flip a coin.

Thanks Harold! You and the others make excellent points! Perhaps we will flip a coin!

We returned last April from Prague, Vienna and Budapest.

I would suggest you just take EasyJet from Gatwick down to Budapest and spend quality time there. Prague was much more expensive (except beer) and weekends brings British revelers in mass. We just preferred the atmosphere in Budapest better, and with 200 museums you never run out of things to see.