When you meet clergy of your own faith, how do you greet each other and what is this greeting meant to signify? How would you greet other religious, such as the dalai lama?
We greet clergy as we greet others, always signifying our unity in ministry. The closer our relationship, the more personal the greeting, although no particular greeting is reserved for clerics. Bonds exist between the priests of a diocese, between those who have studied or ministered together or collaborated on special projects as well. Similar bonds exist between clerics and others also.
I believe there are two factors which influence our types of greetings: culture and circumstance. A handshake along with relevant words would normally accompany any greeting in our western culture. Yet, things can be entirely different for others. Some involve very formal acts such as bowing, embracing and forms of kissing or touching. State occasions also dictate a much different form of greeting.
Then, as you point out in your question, there are the very formal meetings with senior clerics of any faith. Here, one does ones best to follow the proper protocol. For instance, you are correct in your example of meeting the Holy Father. Upon being presented to him, whenever possible, we do genuflect and kiss his ring. This is as much a mark of our respect for the position as it is for his authority; a sign of our obedience to him- something we promise to our bishops at our ordinations. (Different forms of protocol exist for Heads of State and other faiths.) Much of this formality has disappeared from our day-to-day lives here. Nonetheless, we always strive to show the same dispositions to our leaders - although the form often changes.
When we meet each other, let us be inspired by the greetings which began the many formal letters written so long ago by Paul to various communities: "...May grace, mercy and peace be yours from God the Father and Christ Jesus our Lord..." (I Tim: 1:2).