Hey, Alice, that sounds like a great trip! Sadly, I never got to interrail back in my teenage years (and early 20's), and I'm probably too old for it now, but I've been around Europe quite a bit.
I think your itinerary sounds fine as it is. I've never been to Bern, Ljubliana and Venice (although I aim to visit the last two next year on a road trip), but I've been to the other places. Prague, Paris and Budapest are IMO the most beautiful cities in Europe, and I'm glad to see you have Prague on your list. Brussels is fine, but I'd rather take a 30-minute train to Ghent from there (or Brugge) to get some fantastic Medieval architecture, great eating places, pubs etc.
Berlin is a nice city too, but a bit frustrating, as there doesn't seem to be a real city centre with a pedestrian avenue etc. More like several centres spread out across the town, a bit like LA. Hamburg is better that way, but from what I remember a more industrial city (it's been many years since I was there).
Rome is always a treat, but be prepared for LOTS of tourists. The big tourist sites, in particular (Colloseum, St. Peter's etc.). When I was in the Vatican Museum last month (to see the Sistine Chapel), it was like walking in one long exodus from a big football match or something. Not really the cultural mood I was looking for. The prices are also steep, unless you know someone to take you outside the tourist routes. But there's obviously lots of tradition and a ruin or historical site wherever you turn, so lots to see.
Amsterdam is cool. Cozy atmosphere, quite 'beatnik'. But don't go wandering into dark back alleys alone.
Copenhagen is always great. Lots of pubs, restaurants and intimate atmosphere.
And of course, London is a classic. But most pubs close at 23-midnight, and it's not always easy finding a night club for clubbing after that if you're in a pub-heavy area. Check around first.
I wish you an excellent trip, Alice. Travelling is my favourite thing in the world, so I'm quite envious.