Student or Learner
Is med school, dental school, pharmacy school, business school, all considered grad school? If I were applying to pharmacy school but didn't want to be specific when telling acquaintances what kind of program I am applying to, would it be correct to say, "I am applying to grad school"?
Having worked for a company that provided student loans, I can attest that there is a difference between a grad school and professional school. Law, dentistry, medicine, etc. all fall into the latter category. If you are getting an MS, and MA, or a PhD, you are in grad school. However, the majority of the population doesn't always make this distinction.
If your brother is getting his law degree, chances are, your friend won't ask which grad school he's attending, but which law school. If your sister is getting her veterinary degree, your friend would ask where's she's going to vet school, not grad school.
My thoughts are for the US only.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.