The short answer to your question is a definite "Yes".
As Ashwin has pointed out, the music you currently have on your Mac's iTunes is yours. You own it, either by download from iTunes store or by importing from your owned CD, for example.
What you were seeing when you opened the Music app on your iPhone was "Apple Music" which offers you a free three month trial, after which you pay $9.99 or £9.99 per month. I'm pretty sure this is not what you want.
As my colleague, Ashwin, said, connect your iPhone to your Mac via USB - and the USB connection should be to a port on your Mac, not via a hub, otherwise it may not be recognised.
As you have El Capitan, and therefore I presume the latest version of iTunes (12.5.3.17), a tiny telephone-like icon will appear near the top of the window just to the right of the Music, Albums, Apps drop down box.
Click on that. The first time, if it is your first time, you will be taken through a configuration process, the options of which are pretty self explanatory. Make your choices.
Then immediately under the phone icon is Music. Clicking on that allows you to choose what songs, albums, genres, artists, playlists etc you wish to sync. Choose and click sync which is near the bottom on the right.
Hope that helps a bit.
Ian