Most Museum Studies Programs are attached to other Academic Departments, such as Art History, History, Anthropology, etc. If you intend to be a curator, you're best off getting a Ph.D. in your field with a concentration or certificate in Museum Studies. If you're more interested in the archival side of museums, you might consider a Master of Library Sciences (MLS) program. If you're interested in the administrative side of things, an MBA or an MA in Museum Studies can be very helpful. Since you're doing an internship this summer, you should talk to the administrators and curators for advice on how to get into the field - take advantage of their experience and ask them if they have any colleagues at university museums whom you might contact.
I'm an Art History major at St. Louis University, I'm starting graudate school for Art History in the Fall. I worked in an art museum for several years, too. If you're interested in Art History-based Museum Studies Programs, I can recommend quite a few. If you're interested in the archival/library stuff, there's a list of uni's with MLS programs here: http://www.ala.org/ala/accreditation/li ... ectory.htm
If you're in St. Louis and you'd like to get together to chat about grad school stuff, I'd be happy to talk to you. I'm a first-generation college student and I basically went into the process not knowing what to expect or what was expected of me.