The path to getting CDs into your iPhone’s music player is via iTunes on a Mac or PC and then a USB cable...
I thought as much but can't work out how to import my CD music files into iTunes.
Do you have the music files? Or do you just have your compilation CDs?
If you have the music files, in iTunes, go "File" and "Add Folder to Library", then sync your Phone to your iTunes library.
If you are doing this on a PC, then I suggest that you set iTunes to copy the media files into your iTunes library before you do this (in iTunes, go "Edit", "Preferences", "Advanced", and tick "Copy files to iTunes Media folder when adding to library").
If you just have the compilation CDs, then you should be able to rip them in iTunes, but it won't know what they are (because home made CD) so you will have to name them yourself. And once they are in iTunes, again sync your Phone to your iTunes library.