[spindle]

I did this by attending a lot of research seminars until I made friends with professors. The universities I've hung out at have all been fine with anyone attending research seminars. A billion years later I was a professor myself. You just have to be friendly but very unpushy, and patient (and smart! - but probably you've got that covered or you wouldn't even be asking).

I followed this plan twice, in public health (mainly biostatistics) and in philosophy. It probably wouldn't work in genetics or astronomy because they require so much technical knowledge before the seminars make sense.