Both sign up and enroll are valid when applied to courses. You can also say you registered for a class or a course. However, when talking about a school you can only say "enroll." I enrolled in that school" and not "I signed up for that school."

You could sign up to run a race or sign up to volunteer at a fundraiser, but you only enroll in a course or enroll in school. The main difference is that enroll is more formal, and it can mean becoming a student, as opposed to choosing a particular activity or class.

When talking about officially registering as a member of an organization or student in a course, it is better to use enroll. Using the phrase “sign up” sounds fine if you are saying “I signed up for a course”.

"Enrolled in a course" is more formal than "signed up for" a course.