The best advice I can give you is to do more reading first and narrow down what you want to see the most. With so much to see, you will rapidly find 2-3 weeks shrinking, so once you know your priorities, the people here who have visited several times can help you maximize your time and give you all the vital travel tips. At this stage, you are flush with excitement, and a lot of what you hear from us will be shots in the dark.
For the young budget traveler, Let's Go, Lonely Planet, and Rough Guide are all good resources--but that of course does not preclude getting advice here!
So, I can throw out a few more tidbits, like try to fly open jaw, but again the most important thing is to determine your dream trip--what I hope will the the first of a lifetime of travel. I feel like I could go to Italy every year the rest of my life and not see everything, but in my first two trips I have not seen one major tourist attraction in the country (so you see the dream trip of one differs very much from the others).
You have so much time to plan, there is no reason to not just have your nose in books at this point.
2-3 weeks and $2-3000 are bid differences--that does not include the cost of air travel, does it?

2-3 weeks seems like a perfect amount of time to explore Tuscany in some detail. Weather should be nice in May/June and is a preferable time to travel in Italy compared to July and August.

I think there are direct flights to Pisa from JFK, although you may not want to spend too long in Pisa. Look at Lucca, Siena, hilltop villages, Florence etc.

We Fodorites can be great resources, but you need to be doing your own research, as well, to decide which of the many wonderful places in Italy will be the places that you prefer to visit. Start with your local library, hulu or Netflix for travel DVDs and shows, and your local library for travel guides that describe the various places. Lonely Planet guides are more geared for the kind of travel you have in mind, heavy on using public transportation, hostels, etc.

Also, I have heard that Thorn Tree travel forum has more travelers of your type, as well, so that may be a good resource too.

I made a similar trip to Europe after I finished law school, but was able to take a month-long trip, with my mom for half of the time, and my husband for the other half. We stayed in moderate hotels, though, and rented a car for portions of the trip. It was, indeed, a great way to begin my professional life!

I think there are direct flights to Pisa from JFK, although you may not want to spend too long in Pisa. Look at Lucca, Siena, hilltop villages, Florence etc

This is not impossible but his area can be hard to tour on a budget. There are very few hostils in the hill towns and the bus schedule can be difficult.

BTW camping in a tent is a no-no in Tuscany

You're not going to be able to explore all of Italy in 3 weeks so you need to choose your destinations carefully. Determine your interests - what intrigues you about Italy? Food, architecture, history, etc. That will help you narrow down your locations. If you choose cities/towns that are not near each other you need to factor travel time into your total vacation time.

$3,000 for 3 weeks is a good amount of money for food, lodging, transportation, and sightseeing so you should be able to comfortably afford a 3 week trip. To save money you can stay in hostels or convents. There are plenty of inexpensive 1 star hotels.

You mention BRINGING dollars - do not do this. Bring an ATM card and get Euros as you need them.

Read several guide books, this forum, and the thorntree forum for ideas of where to spend your time. I would advise against traveling to lots of places since you won't get a flavor of any of them and will spend extra time and money going from place to place.