As an intern I got to know the full-timers working in the cube-space around me. As my internship term came to an end, a nearby coworker (and friend) suggested I talk to manager (and friend of his) in another division. I interviewed with 2 program managers, and they offered me the position at the end of the interview. It was a relatively comfortable interview since they were looking for someone with my skills. However, unlike HR interviews, the researchers at Argonne just want to know what you can do for them and aren't actively trying to screen you out.
Projects come up quickly at Argonne - meaning that new internal job opportunities do too. So my advice to recent (and soon-to-be) graduates is to land an internship there (relatively easy, just an application) and in the meantime learn about the projects going on around you. Get to know your neighboring coworkers and seek out the interesting aspects of their jobs. Find a project you like, then get to know the lead researcher.
Argonne is a pretty laid back community; they may be brilliant scientists, but they are typically wonderful and interesting people as well. Most of them work to live, not the other way around. If you put yourself out there people will be happy to help you progress your career.