I've given it a bit more thought.

Start in London, the access via Europe (lite) is always an easy start there will be crazy shops selling stuff but the muuseums will be all open and the sights interesting. Also, it can be a comfortable place to get over jet lag

Train to Paris, but you need a spectacular restaurant to see the new year in, it also needs to be close to any appartment you will be renting. That will be an interesting hunt.

Amsterdam after these first two my be a bit of a down grade and unless Rembrandt, diamonds and cake are big on your list, I might cut out, especially as you will have a warmer Rome to get off to. Fly, don't take the train, easy to do via cheepo airlines. Of course you could stop by Nice maybe on the way, but I'd not bother. An alternative is to fly by Florence and then train to Rome. Now Florence is so important. (Venice another stop by but more an out doors sort of city).

Fly out of Rome

suze on Jun 27, 16 at 7:59pm
<Paris first, Rome last for sure>

Because her freind wants to spend New Years there, her dream.