Do Earth's layers move at different speeds?
Im am currently doing my masters in geophysics (last semester) and before that I did a bachelor in geoscience. 

I assume by layers you mean the crust, the mantle and the core. 
These all have different composition and also different densities. But the earth rotates as a whole, not the individual layers, all layers have the same angular velocity. That means they all make one rotation per day. 
These layers are also not the perfect boundaries we like to imagine, but more a change in properties around a finite depth. This depth can even change at different places.