How to put music on iphone 5?

Okay so I just got a new laptop & my father had linux put on it & I can't seem to download iTunes? I don't know why. Any other ways I could put music on my iPhone?
Also my sister has Windows 8 & she has itunes & I tried to use that but it wouldn't sync any of the music she has to my iPhone? Why?

Download directly to your phone from iTunes

Gotta be iTunes, and AFAIK it only runs on OS X or Windows. Is Linux the only OS on your new laptop or is it a dual-boot system, capable of going into windows and linux? ITunes has a number of advantages and disadvantages - for linux users, it's an easy call to "con".

"Also my sister has Windows 8 & she has itunes & I tried to use that but it wouldn't sync any of the music she has to my iPhone? Why? "

I wouldn't know - I wasn't there and you didn't say how you tried. 2 possible explanations:
She has an older version of iTunes - the new iPods and iPhone require iTunes 11; You just did it wrong.

Try enabling your iPhone to be manually managed for music and video. You do that in iTunes when connected to the phone. Once you do that, you should be able to drag and drop the music files from the iTunes window onto the iPod Music tab on the left-side sidebar. If you don't see the sidebar, no problem. Just go into the view menu and select it there.

Use CopyTrans Manager and transfer music to your device.

You can use rhythm box or banshee to do it I think

You must ragister your iPhone first with iTunes by iTunes id. Then sync.

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