It’s not incompatible–you just need mp3 files, which are compatible across the board.
What you tried to do with the CD wouldn’t work with an iPod either. If you look at a CD in a computer that reveals file types, what you’ll see is a list of .cda files. Those are basically the index to the CD–they tell the music player (iTunes, Windows Media Player) where to look on the disc. You put a bunch of tiny 44kb index files on your player–they don’t play. Not a question of compatibility, just the way portable players work.
CDs always need to be converted or ripped–Apple calls it “imported”–so their files can be used in a music player–doPi, Sansa, Sony, your phone, whatever.
If you leave the defaults in iTunes it will rip to .m4a files–Apple’s proprietary files. If you change the iTunes defaults (look for Importing settings, iTunes keeps moving them around) then you can make the default .mp3 at 320 kbps, which is all-around compatible.
If you’re an audiophile you would want to rip to FLAC instead, which would take different software–but assuming you’re using the player for portable music through less than high-end headphones, mp3 at 320 is fine.
If you have an iTunes library in Apple formats, you can convert them to mp3 in iTunes. Or you can get a Clip Zip instead, or geek out and use Rockbox to play the m4a files on the Clip.
If you’re starting with CD’s, you can rip them directly to .mp3.
By the way, I hate the latest iTunes as well. They completely messed up the interface. You can try rolling back to a version you like: