I think what helped me was knowing what kind of research I was interested in and having the background for it. I didn't have any connections, and I didn't email PIs asking if there was space. I wouldn't suggest doing that if you really want to get in (I was actually going to start emailing but I heard back really early), but if you describe your past research or related classes well in your application, it would really help.
Oh and apply early. I think most PIs decide by January/February or something.
Scholarship Access: Becoming a Student Doctor course
