In the Name of Allah the Most Gracious, the Most Merciful

The Muslim’s motto is to take the lead in greeting people:

Almighty Allah says:

(And when you are greeted with a greeting, greet with a better (greeting) than it or return it; surely Allah takes account of all things)

[An-Nisa’: 86]

The Muslim’s motto is to initiate the greeting to people whenever he meets them. He greets both the people he knows and the ones who he does not using the Qur’anic form of greeting. A Muslim refuses to use other forms of greeting which are used by people who are far from Islam to preserve his Islamic moral character.
Allah the Almighty says:

(Their salutation on the day that they meet Him shall be, Peace)

[Al-Ahzab: 44]

The meaning of the word “Salam”:

The word Salam (peace) means As-Salamah (safety) in its absolute sense; the safety of religion, the safety of health, the safety of honor …etc. Thus, when you say to your fellow Muslim brother:
‘Assalamu Alaikom (may the Peace of Allah be upon you,)’ you invoke Allah to grant all kinds of safety to him, starting from the safety of religion and belief, to the safety of health, honor and reputation.
Moreover, As-Salam (The Giver of Peace) is one of the ninety nine Beautiful Names of Almighty Allah, so if you apply His Method, you will be granted the blessing of this Name, and you will live a happy life. Almighty Allah says:

(Whoever does good whether male or female and he is a believer, We will most certainly make him live a happy life)

[An-Nahl: 97]

You will live a happy life in this world, and in the Hereafter you will enter Paradise as wide as are the heavens and the earth.
You are in peace in the worldly life as long as your treatment with Allah is in peace. Furthermore, Allah invites us to the abode of peace, namely, Paradise. Thus, if you enter paradise, you will be in the abode of peace. I told you earlier that this word has a very wide sense.

The obligation of spreading As-salam:

Muawiya bin Suwaid that Al Bara’ bin ‘Azib, may Allah be pleased with them said:

Allah’s Apostle orders us to do seven things and forbade us to do other seven, he orders us to follow the funeral procession. to visit the sick, to accept invitations, to help the oppressed, to fulfill the oaths, to return the greeting and to reply to the sneezer: (saying, “May Allah be merciful on you,” provided the sneezer says, “All the praises are for Allah”…

[Al-Bukhari]

In another Hadith reported by Abu Huraira: The Messenger of Allah, may peace and blessing be upon him, observed:

You shall not enter Paradise so long as you do not affirm belief (in all those things which are the articles of faith) and you will not believe as long as you do not love one another. Should I direct you to a thing which, if you do, will foster love amongst you: (i.e.) give currency to (the practice of paying salutation to one another by saying) as-salamu alaikum

[Muslim]

Habituate yourself to pay salutation to one another. Some of the companions used to pay salutation to one another whenever they met. They even greeted each other again when a tree came between them. Can you believe that?
If you sit in the company of some people and you leave them for something to do, get yourself used to greet the attendants when you come back. Whenever you leave a meeting for some reasons, say Assalamu Alaikum upon your return. These words are pregnant of so many meanings. Let alone, that they are derived from the Beautiful Name of Allah As-Salam. They mean safety and protection in their wider sense.
Abdullah Bin Salam, may Allah be pleased with him, said:

The people hurried quickly to meet the Messenger of Allah (PBUH) when he arrived in Al-Madinah. They cried, ‘The Messenger of Allah has arrived! The Messenger of Allah has arrived!’ I came out along with the people to see him. When his face was clear to me, I realized that his face was not that of a liar. The first thing I heard him saying was, ‘O people, spread Salam (the greeting of peace: peace be upon you), give food, join the ties of the wombs, and pray (at night) while people are sound asleep (negligent), and you will enter Paradise in peace.’

[At-Tirmidhi]