LOL. I HATED AT music when I was first learning. I mean, it was OK for about an hour, but after that I just wanted to listen to anything else. (And then I came home, and as a way of making the experience last longer I listened to almost nothing but the two tango CD's my teacher had made for me. And I got to like it, albeit I also got tired of those particular songs.)
In addition to the beats and rhythm, etc., and the emotions, there's also the sounds of the instruments to pay attention to. (And of course that feeds into the emotion of it.) This is where I become a fan of how my mom taught me to listen to (classical) music--with programmatic stuff. Peter and the Wolf and Carnival of the Animals are fantastic starting points. Start with Carnival...pick one of the pieces...and pay attention to what instruments the composer used, and how he used them, and why he used them. Listen to "Acquarium," and think about how the music is like standing and watching fish swimming smoothly and gracefully through water. Listen to "Kangourous," and think about how the composer used the instruments to paint a picture of how a kangaroo moves. It's all right there. Think about how you would move to music like that.
Move to Peter and the Wolf. Read the story of it, and read about the leitmotifs used and how each instrument represents a character. Then listen to the music, and try to pick out the instruments, and try to identify each leitmotif. Listen for the bassoon (the grandfather) and try to understand why that instrument would be associated with that character...or the bird with the flute...or the hunters with the percussion. Learn to hear all of those various instruments, and how they interact, and why they were chosen.
Then...go back and listen to AT music. Think about how you could move differently when the music is legato versus staccato. Think about the rhythms and counter-rhythms. Feel the shifts in energy, and the ebbs and flows. Feel how some pieces sound wistful, or angry, or romantic...and think about how you can move your body differently. This is all musicality--beyond beats and emotion--and it can all be taught.
(Other good programmatic pieces are The Sorcerer's Apprentice, Rodeo (by Copeland), and Pictures at an Exhibition.)