Transfer all music from old phone to new iPhone?

So I had an iPhone 4s and just got the new iPhone 6 and I want to transfer all of my music to my new phone. Problem is, the only ones I can add so far are the ones I purchased through iTunes. The computer I have all of my music on is pretty old and when I plug the 6 in. It thinks it's a camera. I can't even drag files into it.
The files are everywhere on this computer and it would takes weeks to drag all of them, correctly, onto my laptop (which I should eventually but I have a busy schedule right now). Oh and the old computer's ethernet cord doesn't even reach our router either…
Is there absolutely any way to transfer everything just through Wi-Fi and on the phones? Or at least transfer my entire library onto another computer (with all of the correct files rather than just copying directly from iTunes).
Since I didn't think you could back up your music to iCloud…

I guess this app named AnyTrans will help you transfer all music from iPhone to iPhone, from computer to iPhone or from iPhone to iTunes.See detailed info: http://www.imobie.com/...iphone.htm

You can use iphone transfer software to copy Apps, Contacts, Songs, Videos, Photos, Playlist, Games, etc between iPhones/ipod/ipad/computer.