To be perfectly frank, it sounds like your friend doesn't know shit about Europe.
That trip is going to be more expensive than it's worth, tiring, and you won't know anything about any of the places you visit after you leave. I've lived in Europe for 5 total years and traveled there for visits a couple times as well and here are my tips:
-Pick 2 places, maaaaybe 3 tops with that much time unless you want to see nothing but all the gay shit you can Google without waiting in line with a bunch of other tourists
-Everyone says take trains, but I've often had stunning success with airlines and paid the same or less for a 1 hours flight rather than a 2 day train driven pain in the ass. Keep an open mind, Skyscanner.com is great
-Find something you actually want to do in these places. All those cities are just big capital cities when you get down to it. They all have bars, restaurants, pub crawls, etc. just like every other huge city in the world. "Crazy! EVERYONE here speaks English for some reason!" Not a coincidence... Sure if you have no foreign experience you'll have a great time regardless just due to culture shock, but most of those countries have way more to offer than their political center. If someone asked you about the US would you say "Oh you HAVE to go to Washington DC and see the White House" ? Probably not....
-Why on Earth did you decide you want to go to Milan for a day?
-I forgot to mention buses. Which can be dirt cheap and extremely easy as well.
-If you travel that much you won't be able to party like you think you will. Staying somewhere for more time would benefit that aspect, places like Spain are dead til 2-3 am for example.
-Everyone and their mom chooses the summer to travel to Europe. And they all go to the same 4 cities they know how to pronounce. Maybe consider some places you didn't think of like Croatia, Greece, or Portugal and enjoy the beaches and culture. Croatia for example is super nice, kind of like Italy and Greece mashed together, and it's dirt fucking cheap.
The biggest point I can't stress enough is that as long as you make a decent choice, 6 days in one place is infinitely more valuable, more fun, and more cost effective than 2 days in 3 places.