Transfer music from old computer to new one using iPhone?

I just got my own computer for college and I want to put the songs from my iPhone on my new computer. I don't want to simply transfer the data from the old computer to my laptop because it was a shared computer with many songs that I don't want. I have most of my songs on my new computer by use of iCloud, but my playlists and some albums won't transfer. I have tried plugging my phone into the new computer and backing up, syncing, everything but it hasn't worked. My phone is an iPhone 5 and the computer is a new, current model, MacBook Air.

ITunes won't allow you to transfer music from the iDevice to your computer - only the other way around

To do this, use DiskAid
http://www.digidna.net

Try copying the itunes folder in "my music" on your old computer, to the "my music" folder on your new computer. This contains all playlist data, and such.