What are the Islamic greetings and their proper responses? Is there a special way to greet one another while departing?

Al-salāmu ‘Alaykum (peace be to you) is the only greeting that Islam has laid stress on when people meet people. The Prophet (sws) through his established practice has directed Muslims to use this greeting when they meet one another. This greeting is to be answered in a similar fashion by saying Wa ‘alaykumu’l-salām (and peace be to you too).

In this regard, the correct etiquette is that a person should try to take the initiative instead of waiting for others to begin. The Prophet (sws) is reported to have said:

The young should greet the adult, the pedestrian should greet the stationed and the small group of people should greet the larger group. (Bukhārī, Kitābu’l-Istīdhān)

Secondly, this salutation is not confined to known people; even unknown people one comes across should be greeted by it --since it basically is a prayer of well-being. The Prophet (sws) is reported to have said:

A person asked the Prophet (sws): ‘What is the best in Islam’. The Prophet replied: ‘To feed the poor and to greet a person you know with al-salāmu ‘alaykum as well as to greet him whom you do not know. (Bukhārī, Kitābu’l-Istīdhān)

Using this greeting while departing is optional. One can say anything that expresses good wishes.