Heartstrings (but honestly after the first half it starts to drag, but its worth a look)
Mary Stayed Out All Night (not exactly idol, but main lead is a singer in a rock band and does incorporate music)
What's Up (about the musical world)
Shut Up: Flower Boy Band (only watched first two eps, saving it for summer, but from what I've read its really really good)
That's all I can think of. I know dramas that star idols who play regular people, I also know dramas which don't involve the idol world but focuses on like the acting/celebrity world but the ones I've listed are more focused on the idol world and music world.