Mary began telling about her [...] to the club.
I thought of the word initiation but I think an initiation is more like a "rite." I'm looking for something that just means the process of joining a group or a club.
If there is even a slightly structured joining process you could use the word "enroll" (US) or "enrol" (UK):
to become a member or participant
"Mary talked about enrol[l]ing in the science club".
or, to fit your example:
"Mary began talking about her enrolment..."