I was luckily working under a very attentive mentor - he checked in with me every day, asked about my goals, and related anything we were doing to architecture (my major). Visiting other departments, like the Wood Shop, Conservation, Curatorial, and Archives showed me every branch of the Museum World. I enjoyed exploring restaurants and other museums around DC too.
What I wish was different
If I could do this again, I would make more of an effort to befriend other interns. I was the only intern in my department, so I easily became isolated in terms of office space. I'd invite someone to eat lunch with me out on the town. I'd ask them about their intern projects, visit them at their desk. I think connecting with people going through a similar experience builds a support system and creates longer friendships. I wish I'd spent more time doing that last summer.
Ask questions! Visit other departments like I did. Sit in on meetings. Connect with other employees. You're there not only to gain experience, but also to learn about the industry and if you want to pursue it in the future. If your mentor isn't super-engaging, you can propose an intern research project that benefits the company as well as your personal interests. Find a way to insert your knowledge - being proactive is one of the most attractive characteristics of any job candidate.