Amsterdam-Koln can be done for 19 Euros per person on the train, if you book it far enough in advance on www.bahn.de .
 
Koln -> Leipzig can be done for about 29 or 39 Euros per person on the train, again if you book far enough in advance.
 
Leipzig -> Prague – I don’t know, but I think it’s very cheap based on past posts here.
 
Prague -> Italy – with a budget flight, you could probably do this for 40 Euros per person.
 
Driving around Italy and France - this might be a good use of a car, although I would ditch it before Paris.
 
Paris -> Brugge – 50 Euros per person on Thalys, far enough in advance.
 
Brugge -> Amsterdam – about 30 Euros per person, even if not booked in advance.
 
In short – Eurail might not be a good comparison point for cost. A car may still be a good idea for part of your trip, and perhaps you like driving well enough to fork out the extra money for the rest of your trip, but much of your trip can be done very cheaply without a car.
 
 
 
 

If the principal driver is under-26, there may be a problem with car rental.   Add to that very high prices for Petrol, Insurance (CDW) and Tolls (on toll toads or bridges) and you may not be able to justify car rental with a barebones price comparison. 
 
The ‘Under-26’ that’s a disadvantage for car rental becomes an advantage when you are shopping for RailPasses.  You qualify for ‘youth’ RailPasses.
 
Or, if you’re older than that, you can buy a FlexiPass for two people traveling together, and save a few Euro. 

I prefer the train for basic transportation.  It’s faster, cheaper (unless there are 3 or 4 of you willing to try to fit yourselves and all your gear into a very small subcompact) and more convenient (usually city center to city center).  I also think it’s less stressful — which quickly becomes a big issue in the 24/7 togetherness of travel.

Renting a car is a great way to explore the countryside and makes it logistically possible to fit more sights into a couple of days when you’re not in a city.  Most tourists are interested in the big cities and/or places like Brugges which aren’t vehicle friendly.  A car is a very expensive nightmare in a city.

I just have to add that the rent prices for a car in Europe are way higher than let’s say, in the US. Also, the gas prices are high – they’ve never been low in Europe. And even though Americans are now crying because of skyrocketing prices of their gas, it is still cheaper there than in Europe! Also, consider that European medieval city centres were not built with intention for car use, so there are often long and slow traffic jams or hefty fees for entry into the historical centre. And of course, medieval city centres are lacking parking spaces, so there goes your money again for parking.

But if you want to have total freedom of movement and want to change plans on-the-go, or you are interested in the countryside as well, go for a car.

As others have already said, the costs of car rental in Europe are very very high! I’d recently checked on a one-day rental, and abandoned the idea when I added up all the various costs and fees.
I’d start by thinking about where you want to go, and all the places you’ve mentioned so far can easily be reached by train, so I’d say to use the trains.
In your case, you’ll be in Italy, and possibly having a car for one day in an area such as Tuscany would be worthwhile. One more caveat, though, is that I’ve heard horror stories about car rentals in Italy—- things like extra unexpected charges for pre-existing damage to the car, etc. It wouldn’t be a bad idea to photograph any minor damage that you find on the car at the time that you pick it up, and note those damages on the rental agreement before driving away.
Finally (and I know that your plans currently don’t include France, but for the benefit of others who might go there), look at a site like Bahn.de to help you know where trains do or do not go. You’d be surprised how many very small towns and villages do have rail service—- or bus service from a nearby bigger town. Within France, for example, the Beaujolais region would be hard to see without a car, and there are some very nice villages near Colmar (in Alsace) where having a car is a virtual necessity.

Thanks for your comments, it sounds like car for part of the trip is a good idea outside big cities (eg in Italy, France), and the train for the rest. I’ll try convince my boyfriend!!
 
 

Eh, one more thing…

I have never been to Australia, so I don’t know: Do your cars use automatic gear shifters, or manual ones? Because if you are used to drive around with an automatic gear shift, it may be very difficult for you to get used to operating the manual gear shifter. Almost all European cars have a manual. Watched many an American struggle with that here in Europe []

 
Drive, you’ll save much money & time.  Try to stay outside of the centres and try to park near an outerlaying metro stop.  European cities can take some getting used to with the roundabouts and all, but on the highway it’s all good.

Hi ‘Papyr’ – we have manuals here! The big differences are - everything’s on the opposite side of the car, and we’ll be driving on the other side of the road!!!!
I’m sure we’ll adjust
Thanks also ‘heavydrinker’ – parking outside a big city will probably save parking fees and we can just catch a train into the city!