What is the exact process one must go through in order to serve at a monastery?
Speaking as a monk, I can say that there is no exact process. Monks and nuns come to the monastic life through an amazing variety of ways.
Firstly, since we live in an information age, start reading up on the monastic life, both theoretically and in the lives of monks and nuns.
Secondly, spend time at an monastery. The process and the possible vocation will unfold and start to clarify with such a visit/s.
In my day and age we were always recommended to read Ignaty Brianchaninov's "The Arena."
It is good both for an enquirer, to get a grasp of monastic principles, and it remains a very good book to consult after you have become a monk.