So, there are no cut-and-dried paths in museums. There are a few basic rules of thumb - if you want to work in archives, you need an archive-specific degree; if you want to be a curator in a really big museum, you almost definitely need a PhD in a subfield - but between the fact that every museum has its own system of roles and hierarchies and that the job market for museum professionals is terrible now, it's tough to know what to tell you. Technically, these are all different things that might be done by people in different positions, except in quite small museums where everyone has to do everything. Look at the staff list at the Albany Institute of History and Art: for outreach you have a director of community engagement, exhibition design is done by the curatorial staff and the exhibition designer, and the directors of programs and education assistant handle hands-on education. (By contrast, in my museum I do all three as "collections manager".)

The basic advice I'd give you is to get into a good museum graduate program, like Cooperstown, focus on these things while you're there, try to get internships specifically dealing with those duties, and then get a job in it. Training for specialization in museums frequently has to do with what you decide to make of more generalized positions. But ... the job market is terrible, and I can't recommend getting a degree in this field unless you want to do it with every fiber of your being. I've gone through some pretty dark times over the last four or five years trying to get jobs or trying to deal with the jobs I have.