Cheapest might be bus..but its the slowest and while you think 4 weeks is a long time it will go fast! A week in Amsterdam seems way too long as its not a large city and 2 days would be ample there…I had just under 2 weeks in Italy and that was only central/northern Italy ( verona, venice (2nights), Sienna, Rome (3) Florence (3) and Bologna and found I still needed more time for some places and missed out on other places I wanted to go. It may seem small but there is a lot packed into Europe!
Anyway for buses, which sometimes have cheap bus trips try. I travelled from London to Dublin for around 16 pounds..that was a 12hr bus ride. If you get the night ones you can sleep on the bus and save acccomodation and save on travel time ( which when you are travelling around a lot like you are will become annoying at times as all you want to do is GET there and explore.)

There are also ferrys I think from Italy …say around Bari which I know go to Croatia ( where I went to and it was lovely) but also go to Greece ( where I never went) and might be an option, but I can’t remember the name..just do a search on the net.

Anyway cobber have a great time..try to focus on some areas and explore them well which will increase your enjoyment of an area, see it better, remember it better ( than a fleeting view you experience it) and save on the relentless travel. Italy is wonderful, UK has some neat things, Amsterdam is small and I never saw Greece….
cheers
kiwi traveller

But…hey..its a long way…

And I guess you would get a bit tired.