Hello Carolita
Depends on what kind of degree we're talking about. In Spanish we have different names for them:
Licenciatura
Maestria (or Master)
Doctorado
Also, High School could be "Escuela secundaria". Bachillerato is one of the many different options within the high school. Of course if you just say "bachillerato" everybody would understand what you meant to say.
Good Luck!
Noelia
http://www.nativetuition.com