I checked the testquestion you mentioned but I think you misread it. All the information is there:
Voy a cambiar a mi hijo de colegio si no mejora pronto.
I will transfer my son to another school if he doesn't improve soon.
In Spanish we use "cambiar de colegio" to refer to "transferring from a school to another" - there is more text in the English but in the Spanish that shorter expression implies all of it.
Another similar sentence could be for example:
En el hotel nos cambiaron de habitación porque en la nuestra había cucarachas.
At the hotel they transferred us to another room because there were some cockroaches in ours.
We use the same for example to "get changed" (clothes):
Tengo que cambiarme de camiseta. Esta está sucia.
I have to change t-shirt. This one is dirty.
Cámbiate de zapatos. Necesitas unos más cómodos.
Change shoes. You need some comfy ones.
I hope this clarified it.
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