Do graduates from Zicklin School of Business at Baruch College go to Harvard Business School?
HBS accepts students from a wide variety of colleges. Your resume (college GPA and involvement, post-college work experience), strength of app (test scores, personal statement and recs) and interview will determine whether you get accepted, not the name of your undergrad.
Harvard MBA program generally requires 2-4 years of work experience before you apply. Even more than other top MBA programs, they are looking for applicants that will bring something unique to the classroom discussion ( they don’t want a class full of investment bankers and consultants although there will be many of those type of students). I don’t think you can “expect” to go to Harvard for your MBA, you should do something interesting with your career and decide whether going back for an MBA makes sense to apply. I know first hand Harvard MBA graduates that went through the SUNY system for UG so it is possible, just not probable like any highly selective institution.
Also, although it is heresy in some parts, HBS is not necessarily the best place for any given person to get their MBA. One of the best things about working for a few years first is that it gives you a better perspective as to why you want an MBA (or similar), and what you want to get from the experience.
The Harvard MBA is seen by many as the ne plus ultra of MBA programs, but I know people who have turned it down for other programs that suit their goals better.