So what's the big secret? Get to know as many people as possible.
Let's be honest, being an intern can be scary. You're trying really hard to impress and to be perfect and it's a lot easier to be perfect if you just keep to yourself and hide at your desk. You're not doing yourself any favors. You should be proactive about talking to, meeting with, and learning from as many people as possible both in and out of your department.
In general, you should try to participate in office social events, talk to the people you sit near, and try to make sure you know who everyone is and what they do (of course that will be a lot harder in a big office, but make sure you at least know your department and anyone that your department works with directly as well as major decision makers like vice presidents). But the easiest way to build relationships is to take advantage of your internship to set up meetings with as many people as possible. Here's how:
1) Express an interest in learning more about what their work
If you're lucky, sometime during your first week your boss will walk you around the office to introduce you to everyone or have everyone welcome you during a meeting or via email. Take advantage of that moment. For example, if your boss introduces you to the team during a meeting, take a moment to say something like "thank you, I'm excited to be here and look forward to learning from all of you." If you get introduced individually you can be a little more direct saying "you work sounds interesting, I'd love to find a time to talk to your and learn more about that while I'm here."
2) Ask for some time to talk
Your fellow coworkers are busy trying to meet deadlines and get more work done than they probably have time for so don't expect them to drop everything to listen to you talk about your dreams over Starbucks. But, you have an advantage and you should take advantage of it: most people are willing to help students. Put together a concise, respectful email asking if you could have 15 - 20 minutes of their time to learn more about their work and the path they took to get there. There's a good chance most people will say yes. If you have access to their calendar, take a look beforehand and suggest a few times that you are both available.
3) Follow up
When someone says they are willing to talk to you, don't just let that slide. I've had a few interns ask to talk to me sometime but they never followed up to actually schedule something. You know when your internship ends and you have to be responsible for making the most of your experience. So, if someone tells you they are willing to take some time to meet with you don't let that slide.
4) Have questions ready
If someone is willing to meet with you, be respectful of their time and have something prepared to talk about. For ideas check out this post on informational interviews.
5) Follow up
Yes, more follow up (always follow up). Be sure that you send each person a thank you email after you meet them. Mention something that they said that you found particualrly helpful or interesting or something that you plan to act on (for example you downloaded a book they recommended). Be sure to check in with them throughout the course of your internship and continue to follow up even after you leave.
By getting to know as many people as possible duing your internship, you'll acheive three goals that are the markers of any successful internship: understand how the organization works, learn about different career paths, and build a bigger network.