As many others have said, using iTunes is the simplest way. All you have to do is go to iTune, File > Add To Library > Then choose the music files you want to add. Then just sync your iPhone, and boom.
Although, it you really want to avoid iTunes outright, which I understand if you do. I have had countless problems with their apple id system with authorization, and iCloud account validation. Sometimes, you just want it to work. And that is why, there are many 3rd party programs out there that will help you achieve your goal. Although I haven't used it myself, I hear that MediaMonkey(
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An, iTunes free iPhone/iPad file manager I have had experience with is CopyTrans Manager, which I had very good experiences with, and highly recommend it. And did I mention that it's free!
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