ITunes library on more than one computer? New Mac

Okay, so I'm pretty dumb tech-wise, folks, so bear with me please haha. Right now, my iTunes library is on a neighbor's computer because I didn't have a computer working at the time. So now, every time I have to update my iPhone or sync it or whatever, I have to go to her house. That was fine, but now I'm going off to college, and getting a new Mac laptop.

I was told that I'd have to go to my friend's computer and save all my songs on a USB or something and then transfer them to my new computer. But I've also heard I can just authorize my new computer and plug in my iPhone, and save everything on that iTunes library.
So I guess my question is, will I be able to just plug my iPhone in to this new Mac and authorize that computer, and not have to go to my neighbor's computer?

Here's a "how-to" on all the Apple recommended ways to do that:

http://support.apple.com/.../kb/HT4527

First of all, I think it was very thoughtful of your neighbor to allow you to use their computer for your music.
Now… I see two possible issues here. If any of the music was purchased through Apple iTunes, is the Apple account, yours or your neighbor's? If not your account that would be the first issue as with your own computer you will need to establish your own login/account with Apple.

Secondly, if any of the music within iTunes on your neighbor's computer were NOT purchased through iTunes transferring to your iPhone will NOT put that music on your computer when using transfer, as the iTunes app only transfers music purchased through iTunes and if there was an iTunes account used for the music… Back to the first issue, whose account?

WIthout specific information for the above 2 issues, but assuming worse case scenarios, meaning not your iTunes account on neighbor's computer AND not all music purchased through iTunes, going to your neighbor's computer is a requirement to have your music available for the new Mac, I would recommend using iTunes backup if that is the only option available.