Usual form "Voy a lavarme las manos".
"Voy a lavar mis manos" comes from "Voy a lavar mi ropa", "Voy a lavar mi coche", and then "Voy a lavar mis manos". You're making no-personal your body.
It's better the first option. The second one it's not so common, and can mean complicate things (like that somebody whashes your hands for you).
It also can mean that you see your hands as a tools, not as body.
Cleaning your shoes could be:
"Voy a limpiarme los zapatos" You're doing it
"Voy a que me limpien los zapatos" You're going to the boots
"Voy a limpiar mis zapatos" Could be both of them. Unclear.
Hope this helps
