Can I get all my music from my iPhone onto iTunes on my new laptop?

I just got a new laptop and I want to get my iTunes running again, I do download a lot of my music on youtube to mp3 though, and I was told by a friend that I won't be able to sync my music onto my iTunes account onto my new laptop because I didn't pay for it. Is that true?

Yep, sadly that is true. Unless you actually pay for the iTunes music, then it won't transfer over through your account and if you sync the two together, it'll replace your iPhone's music with whatever's on iTunes on your laptop.

No!
You could try some free tools to assist you to finish transferring music. Try iexplore, iFunx, PhonTunes etc…
It could help you to copy songs from iOS devices to PC, some of them are free or have trail version.
Hope it could help you