mover vs mudarse
'Move' in English has two meanings. Mover is to shift something. Mudarse is to move house.
mover
To move, shift or stir something or yourself — physical movement.
Mueve el coche, por favor.
Move the car, please.
Moving an object. Moverse = to move oneself.
No te muevas.
Don't move.
Reflexive. Stay still.
Me mueve la curiosidad.
Curiosity drives me.
Figurative: motivation. Mover = to motivate.
mudarse
To move house, city or country — change where you live.
Nos mudamos a Valencia en enero.
We moved to Valencia in January.
Always reflexive for changing homes. Never 'movimos a Valencia'.
Se mudó de casa.
He moved house.
Mudarse de casa/piso = move house/flat.
¿Cuándo te mudas?
When are you moving?
Also 'la mudanza' = removal, the noun for moving house.
Tricky bit
Using 'moverse' for relocating gives a different meaning: 'me muevo mucho por trabajo' = I travel a lot for work. If you mean a new home, use mudarse.