There is a Yamaha shop at Ramada signal that sells Yamaha pianos, keyboards, drums and electronic equipment (and some other stuff, not really sure exactly what).

It's located between East 51 and Jarir at Ramada signal.

There is also a music shop that sells instruments in the old souqs area. The easiest way to describe the location is...

Come down the corniche from City Center heading towards the airport. Go all the way past the Diwan and go through the lights at Bank street. You'll then come to a roundabout with the new QNB building on your right. Take a right and drive up to the next round about (or two, can't remember) and there's a musical instrument shop located here. It sells guitars, pianos, keyboards, drum kits... the whole deal. It's also quite cheap.

I think there used to be a music shop on electricity street, close to The Garden, but I can't say for sure if it's still there.