Cold called a mentor working in an area that interested me and made an appointment to talk. Talked about my background and my future goals, said I wanted to learn as much as possible and asked if there were any projects I could get involved with and really take some ownership of. There was one project that nobody else had really jumped at yet, so I jumped on it and started turning things around quickly. I asked a lot of questions and made a lot of mistakes, but pressed on and worked hard and eventually turned out some quality work. The first paper was definitely the hardest one! (much like learning your first second language)
Once you start to show you'll get work done, people will start offering you more opportunities. I don't think I've met someone in academic medicine who wouldn't love to have a hard working, motivated student churning out papers for them! Most people in academics really like mentoring students too. With each project, I tried to get things done as well and as fast as I could while still maintaining grades/boards (always the most important thing, however sad that may be). Taking initiative is huge: if someone mentions a project, turn around something within 48 hours that shows genuine progress (lit review/IRB protocol ready for editing, build a database, etc.). For the first manuscript, I just tried to write something. Anything. I knew it would be bad, but it's easier to start working with something that's actually written. Even now, just getting something on paper, even if it's not great, is huge. I've learned a ton through the editing process and continue to improve my writing with each new project.
Sorry that's kind of jumbled, but basically: find mentor. Find project. Turn around something tangible ASAP. Repeat. I've had incredible success with cold calls/emails. I'll share my secret for cold calls that you can use to get anyone on the phone no matter how high up they are. This works for C-level corporate officers, academic leaders -- anyone who is a powerful high-achieving person. It's this: call their office number early. 7:00am-7:30 seems to be a sweet spot, though earlier has worked too. Most bigtime PIs, chairmen, CEOs, etc., get to their offices early before anyone else does, especially their assistants who screen their calls, and they are usually fresh and not stressed out yet. Be professional, concise, and willing to take 'no' for an answer, but I've almost always had success with this. Emails can get ignored, but a (brief) call can get things done.