That's very nice to hear.
If you travel several legs, think about an Eurail pass: http://www.raildude.com/en/eurail
Otherwise point to point tickets to special prices are often available, if you know your exact travel date and you book early.
You can book for example at http://bahn.raildude.com
first the leg Copenhagen-Munich and then as second Munich-Rome. Both are available as e-tickets and at this site you get the cheapest fares as it is based on the website of Deutsche Bahn (German railway) which is running the trains you will use.
And if you have future questions do not hesitate to ask.
Have a good time and best wishes, Peter