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I think you’ll want to look for some cheap flights. Prague to Rome and dubrovnik to Athens will be gruelling train or train/ferry stints. If you really want to avoid flights or to take ferries, I’d change the order and fly from Prague to Dubrovnik then ferry/train to Rome and train/ferry/train or bus to Athens. Ferry from Dubrovnik to Athens will involve overnight ferry to Italy and then another overnight ferry to Greece.

I’ve been learning that flights can be much cheaper than getting a eurorail pass. However, I do want to train it for the view and experience. Where would you recommend the best train ride to be according to my itinerary (I will probably switch stuff around)?

Before you decide on flights make sure you figure in all of the costs in terms of both money and time. Many flights start off cheap, but when you add in costs of getting to and from airports, baggage fees, check-in fees, fees for paying with a credit card, etc. flights aren’t always as cheap as they seem. Train is much more relaxing, so unless flying saves you significant time and money train is often better. I’m not against flying sometimes, just pointing some things out. Flight or train for London to Amsterdam is a bit of a toss-up, in my opinion. Paris to Geneva should have some decent scenery for at least part of the journey, but are you going to Geneva for a particular reason?

I think you’ll want to take the train for a large portion of this trip. I’d take the train for trips under 5 hours, use the cheapest mode (all costs considered as Augustin notes) for legs where the train ride is 5 to 7 hours, and fly for anything where the train ride is longer than 7 hours unless a flight was prohibitively expensive. Train is, IMO, a far better “experience” than flying. There aren’t many train trips where the “view from the window” is that great.