See as much of Europe as possible in a week [closed]

As Relaxed said, taking the train is probably a good idea.

If you don't mind which order you visit the cities in, there is a sleeper train from Franfurt to Prague, it leaves at 00.54 so you could leave on the day that you finish work and arrive in Prague the morning after, advance tickets seem to be available at €49 in a seat or €60 in a bunk bed.

From Prague you could take the sleeper train back across Germany to either Amsterdam or Paris (departing 17.15 for Paris or 18.30 for either, both require a change and I think you might have to book the two parts separately).

Regular daytime trains are probably best for the other sections of the trip.

To save money on doing this, book it as soon as possible, to get the cheapest fares.

It's very rushed as there are many things to see in these cities and transportation will cost you dearly relative to the time spent in each place but obviously I can also see why you would want to do that and I think it should be doable.

Because you have so little time, the obvious choice between Frankfurt and Paris or Paris and Amsterdam will be the train. It's not the most economical means of travel but certainly fast and comfortable and it will maximize your time in each city. Take an early morning train and you will even have time for sightseeing before lunch (you can take a nap in the train!).

From Amsterdam to Prague, you could fly or perhaps take a late afternoon train to Cologne and then a night train to Prague. The train journey is not particularly fast or convenient but you save a night's accommodation and avoid wasting precious daytime hours on transportation.

Connections between Prague and Frankfurt are not particularly good so the train does not have any major advantage on this route and you will lose a day to return to Frankfurt in any case. Depending on your budget and preferences, plane, trains, buses or perhaps a night train again could all be reasonable options.

For Rome, flying is really the only option, it's too far out of the way to use trains or buses when time is of the essence. That also means dealing the hassle of going to the airport, security, etc. twice. You'll have to check if you can find cheap tickets for flights with a convenient time and decide accordingly but it will obviously be extremely tight.