There are not fast day trains. That train connection by day takes 8 - 10 hours and 2-3 connections. I also don't think you'll find any very cheap flights, but try www.skyscanner.com.
I have taken night trains in the region, and one nice feature is that they are cheaper than routes further west, making a private cabin very affordable for 2 or 3 people. What's most likely to keep me from getting a full night's sleep is noise from outside (brakes, car connections, signal whistles, etc). Lights or motion could also be an issue for some people.
How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it. This link does not sell most tickets outside of Germany, but is pretty comprehensive for schedules.