I actually went with one friend (whose mom paid for his share lol) but we split up a lot so it was kind of on our own and I had planned the whole thing and he decided to tag along.
Since I'm a student, I got SUPER cheap flights home from Bergen, Norway ($230-$300? I forget - flights to the US out of Norway/Sweden tend to be cheaper I believe) and my flight into London was somehow only ~$500. Look into Skiplagged for cheap flights!
I wanted to go to London, Scotland, Paris, and Amsterdam but I was super set on saving money so nixed Scotland and selected Norway because it was cheap to fly out of. HOWEVER, Norway is SOOO expensive. I didn't realize it even though I had heard it. Like a plain black coffee was $5USD. A Subway footlong meal was $17USD!!! We made a mistake of taking a 15 minute cab ride from the airport to the AirBnb apartment (buses stopped after 10pm or something which we didn't realize) and it was $200USD. Bad.
I live in Manhattan so I thought it would be on par with my cost of living here but no way. London and Paris were on par with my CoL. Amsterdam was cheaper, if I remember correctly.
I booked through AirBnb mostly. Hostels were expensive. I had a hotel in Amsterdam. DON'T BE AFRAID TO BOOK HOUSING OUTSIDE OF THE CITY. It saves SOO much money and it's a 15-20 minute bus ride sometimes but it's worth it. Also don't be afraid to ask strangers for help. I speak French so it was easy, but Amsterdam and Norway were pretty scary (bus announcements e.g. "This stop is ___" were tricky). Be careful because the Parisians can be kind of rude and less likely to help. :p
I also booked a lot of travel through overnight buses/trains to save on housing. We arrived in this horrible freezing rain in Amsterdam at 6am and our hotel wouldn't let us check in. We were exhausted and soaked and it was pretty horrible. Still worth saving the money though haha. Here was my itinerary.
NYC --> London (~$500 flight, ~$200 AirBnB for 4 nights) London --> Paris (~$80? train/"chunnel", ~$200 AirBnB for 4 nights) Paris --> Amsterdam (~$30 overnight Megabus, $70 hotel for 3 nights) Amsterdam --> Bergen (~$80 flight, ~$150 AirBnB for 2 nights) Bergen --> NYC ($250 flight)
Total: $1560
I allotted $500 for spending on gifts and take-home stuff and fancy meals. We had one "nice meal" in every city that was pretty reasonable (except Norway - fuck Norway. I got pasta at a 7/11 and it was $11. A Ramen cup was $4). We also got "City Passes" for London and Paris because we LOVE museums and that saved us a ton. We did do museums in Amsterdam and Norway but not so much to warrant buying the passes.
Sorry this is so long! But it was such a great dream of mine that came true so I'd love to share. Oh! If you're a student, check out STAtravel!