4500$ for 16 days?!!! 4500$ can keep you going for 2 months!
Start reading about Europe…Lonely Planet, Frommers are good sources.
The cities you mentioned are all worth a visit, except Milan which I’d replace with Florence or Venice.

Doing it yourself will cost you less than a Contiki tour, but in your case I’d recommend a Contiki tour, as you seem to not know much about backpacking.
A Contiki will save you the planning , but on the downside it’s very rushed and not flexible. But it’s still a good idea for a first trip.

4500$ are more than enough for a 16 days trip.
Get the Lonely Planet: Europe on a shoestring. It will give you tips about budgetting. Browse this website forums, it’s full of good tips. 
Good planning leads to a perfect trip!

London, Paris, Rome are the best bets for a first timer.

In my opinion:

I’ve gone on quite a few trips to Europe and I have to say that $4500 is plenty of money:

Plane ticket: ~$1200
EuroRail: ~$500
Hostels: $30 * 17 = $500
Food: $30 * 17 = $500
Attractions: $30 * 17 = $500
Other: $200

Total: $3400

Then again it depends on if you are planning on staying in hotels, eating out for every meal, or doing other things during your trip that would make it more expensive. Also, organized tours are more expensive and less “fun” than doing it on your own. When you organize your own itinerary it makes it more adventurous, you can skip places that don’t interest you, and obviously cheaper.

Getting lost and finding your way if half the fun.  If you want to be surrounded by other americans the whole time, then go Contiki. 

backpacking = freedom
contiki = senior bus tour
 
 

There are advantages and disadvantages to both. I went with my college (not studying abroad) for a week in Italy and we booked with a travel company for students. It was nice because you get the travel guide that can help with questions and explain things and give you stories you wouldn’t be able to get if you went on your own. However, you are constantly on the go and you have very little time to explore on your own. A year later I went back with a friend and we backpacked on our own. I liked that much better. You ahve ythe freedom to do what you want when you want. You can still take little tours in each city you go.

Plenty of money, but like others said, you should actually read about Europe first. Why did you pick the places you chose? They are primarily capitals and all big cities. That’s fine, but you’ll see a lot more of what a country really has to offer in small towns. It’s a good idea to find out about places, and figure out what actually interests you before worrying about all the details. IMO, trips shouldn’t be about what you can fit in, they should be about what you want to see.

Ah, but it is so hard to photoshop yourself in front of it and make it look realistic.

Some people just like to brag.