apply to: be applicable or relevant
This is the idiom to use when you are putting yourself forward as a candidate for something such as a course of study, or a job. You apply to graduate school. You apply to a company for employment. You apply to a bank’s loan department for a loan.
This is the expression to use if your intention is to obtain something. You apply for scholarship money. You apply for admission. You apply for a job.
The word with in this idiom implies agency, the means “by which” you apply. You can apply with the click of a mouse. You can apply with a printed application. You can apply with a program like Compass, or a service like Monster.