Talk with people in different departments as well as clients and vendors. Read everything you can get your hands on. You won't find everything you need to know in the training manual. Reading contracts, letters, memos, press releases, and trade publications help you become informed on all elements of the organization. Don't gripe about the grunt work.
All internships have some and there is always something to learn by doing it.
Everything has a purpose, so learn how the small tasks fit into the big picture. However, don't be afraid to ask for more responsibility. Milk the fact that you are a student. As a student you aren't threatening and can have more access and opportunities than a full time employee. Everyone wants to help a student learn.