It is probably cheapest to fly to London then train to either Amsterdam or Paris. Once you are in Europe, I would suggest avoiding taking planes. Trains in Europe are fairly efficient. Someone mention railpass already so check out some sites that sell them. I think there is one that is for two people (not 100% sure). One trick to save hotel/hostel fees is to take overnight trains for the long distance trip, say Paris to Rome, or Rome to Amsterdam.
Check weather, it will proabably be warm in Rome by springtime while it's still chilly in London.
I've only been to Amsterdam and Paris myself.
For Amsterdam, must see the Red Light District (drool), Canal tours, the Heinikli museum, Van Gogh (only if you are REALLY interested), etc
Paris, Talamasca said it already, the women are hot (drool again). well Paris is pretty standard for must see. The Arc de Triomphe around 6pm
Goto supermarket and make your own lunch/dinner.
Some hostel actually has private room.
ISIC card is a must
Some museums are free. Likewise for other form of enterntainment
Carry loose change for toilet or go to museums for toilet.
Museum pass is worth it in some case.
Learn the local language (at least "Hi", "Can you speak English", "Thank")