I agree that it's better to spend more time in Paris and Rome and less in Amsterdam. If you're really into the "goodies" that you can get in Amsterdam, you might want a bit more time there, but even in that case I would say 3-4 days max. If you aren't a huge party person, you can fully enjoy Amsterdam in two days. If you're athletic at all, I highly recommend waking up early and going for a run, walk or jog along the water. It's very scenic and Amsterdam has very charming streets. Watching the city wake up truly has an old-world Western Europe feel.
If you're planning on taking a flight from Rome to Paris, I'd recommend Ryan Air. I flew Ryan Air from Rome to Paris last summer. I actually flew into Belleville instead of Charles De Gaulle to save cost. Belleville is outside of Paris, but for around $7 euros you can take a bus into central Paris. The bus stops right in front of the Belleville airport. The bus comes very often and you don't need to buy a ticket ahead of time. This is a bit inconvenient though, so if you aren't super concerned about savings, fly directly into Paris. In Paris I stayed at the MIJE for a week (after that I was doing a study abroad program, so I lived in the Quartier de Latin). The MIJE (Maison Internationale Jeune Etudiantes. I think.)was clean, inexpensive, and overall it felt very safe. Itís one of the better hostels Iíve stayed in.
In Paris I'd say check out the Quartier de Latin. It attracts a very young crowd so its quite fun. Les Deux Maggots coffee shop is quite famous and it can be fun to splurge on a cup of hot cocoa there. For a cheap but very fun dinner, go to the grocery store, buy cheese and a two-euro bottle of wine (the cheap stuff is actually quite good in France), then stop by the bakery and get a fresh loaf of French bread. Sit along the Seine and people-watch with your picnic. Cheesy? A bit. But it's really fun, whether solo or with a group of friends. In general you donít need to spend a lot of money on food to enjoy Paris. I was in Paris two months and I kept to a $10 per day for food budget (yes, breakfast, lunch and dinner). Granted this doesnít work for fancy French restaurants, but honestly, splurge on one or two fine dining dinners out, and that will really be enough. French food is very rich, so a fine dining experience on a nightly basis would be too much in all honestly. For when you do decide to splurge, go to a fondue restaurant with a group of friends. Itís a blast. For day-to-day food basics, discover how many amazing bakeries there are Ė croissants and French bread are delicious and great staples. Crepes are also amazing, and very cheap.
If you are an art lover, youíll be in heaven in Paris. The Louvre, Musee DíOrsay, and the Pompidou are just a few that come to mind. Often neglected is the Musee Rodin. I would say this is a must-go. Not only does it have the most extensive Rodin collection Iíve ever seen, but also the museum itself is a charming building with lovely grounds. But if you donít like sculpture, pass on this one since thatís mostly all there is (obviously). If you really want to see the Louvre, youíll need more than one day. In which case itís a better value to get a student Louvre card. Go down one level below the Louvre main ticket counter and you can get a student pass (I believe itís $15, Iím not sure), for unlimited visits for one full calendar year. If you have an International Student Card you can sometimes talk them in to giving you the student pass for fee (I did hehe).
The ballet is a total splurge, but Iíd say well worth it. Sometimes you can get student ticket rates if you call the box office. Comedie Francaise is a great theatre experience (but you should really understand at least basic French to enjoy it).
Oh wow I just realized this post is getting really long and I havenít even started to talk about Rome yet. If you have any specific questions feel free to ask. Iím very passionate about Rome and Paris in case you couldnít tell. I spent a month in Rome and two months in Paris last summer, so pretty much everything is still fresh in my mind. Bon journee!