Word/phrase for "process of joining a group or a club"
She was inducted into the club.
She was initiated into the club.
The opposite in meaning would be...
She was blackballed by the 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..."