One of the good proprieties of Islam is spreading of greetings because Salam is the greeting of a believer, its motto, and the messenger of brotherhood, intimacy, and love among Muslims. Moreover, it is a good and blessed greeting as the Lord of all the Worlds described in His Saying (Glory be to Him): "But when you enter the houses, greet one another with a greeting from Allâh (i.e. say: As-Salâmu 'Alaikum - peace be on you), blessed and good." [Surat An-Nur: 61].
Furthermore, Salam is the greeting of the inhabitants of Paradise with which angels greet them when the Paradise dwellers are driven to Paradise in groups and the eight gates of Paradise are opened for them, the keepers of Paradise shall meet them with this greeting:"Salâmun 'Alaikum (peace be upon you)! You have done well, so enter here to abide therein." [Surat Az-Zumar: 73]. Salam also is the greeting of the Paradise dwellers among themselves as Allah (may He be Exalted) says: "And Salâm (peace, safety from evil) will be their greetings therein (Paradise)!" [Surat Abraham: 23]. It is also the greeting of angels and the greeting of Adam and his offspring.
It was reported in Sahih Al Bukhari and Sahih Muslim on the authority of Abu Hurayrah (may Allah be pleased with him) who narrated that the Prophet (peace be upon him) said: "Allah created Adam in his complete shape and form (directly), sixty cubits (about 30 meters) in height. When He created him, He said (to him): "Go and greet that group of angels sitting there, and listen what they will say in reply to you, for that will be your greeting and the greeting of your offspring." Adam (went and) said: 'As-Salamu `Alaykum (Peace be upon you).' They replied: 'As-Salamu-`Alayka wa Rahmatullah (Peace and Allah's Mercy be on you). So, they added 'Wa Rahmatullah'. The Prophet (peace be upon him) added: 'So, whoever will enter Paradise, will be of the shape and form of Adam. Since then the creation of Adam's (offspring) (i.e. stature of human beings is being diminished continuously) to the present time." [Sahih Al Bukhari No. 6227 and Sahih Muslim No. 2841].
Of the virtues of Salam is: It is one of the good characteristics of Islam. It was reported in Sahih Al Bukhari and Sahih Muslim on the authority of `Abdullah ibn `Umar (may Allah be pleased with him) who said: "A man asked the Prophet (peace be upon him): What sort of deeds or (what qualities of) Islam are good? He said: To feed (the poor), and greet those whom you know and those whom you do not Know." [Sahih Al Bukhari No. 28 and Sahih Muslim No. 39].
It is the right for a Muslim on his Muslim brother because of the Prophet's saying: "There are six rights of a Muslim upon a Muslim." And mentioned of them: "And when you meet him, say salam to him." [Has been previously attributed to its sources]. It is a great for spreading intimacy among Muslims and to spread love among the believers as the Prophet (peace be upon him) said: "You will not enter Paradise until you believe, and you shall not believe until you love one another. May I inform you of something, if you do, you love each other? Spread salutations among yourselves." [Sahih Muslim No. 54].
Love here is achieved because both invoke Allah for the other with safety from evils and for mercy that brings about goodness. Therefore, it was authentically reported in Al Musnad and others that the Prophet (peace be upon him) said: "Spread Salam among people and you will be safe." [Al Musnad (4/ 286), and Al Albany graded it as good in Sahih Al Jami` No. 1087].
The meaning is: You shall be safe of anything that may scatter you or cut the ties of kinship, so what if that is correlated by smiley face, warm welcome, and good manner?!
A Muslim should salute back with a better salutation or at least of the like because of Allah's Saying: "When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally." [Surat An-Nisa': 86].
The best of two men is the one who starts his Muslim brother with Salam. It was reported in Sunan Abu Dawud on the authority of Abu Umamah (may Allah be pleased with him) who said that the Messenger of Allah (peace be upon him) said: "The person nearest to Allah is one who is the first to offer greeting." [Sunan Abu Dawud No. 5197 and Al Albany (may Allah bestow mercy on his soul) graded it as authentic in Sahih At-Targhib wat-Tarhib No. 2703].
If the one who should start with Salam did not say it, the other should start with Salam and he should not miss the Sunnah.
It is part of the Sunnah that the young would greet the elder, the small group should greet the large group, the riding should greet the walking, and the walking should greet the sitting. It was reported in the Hadith of Abu Hurayrah (may Allah be pleased with him) who said that the Messenger of Allah (peace be upon him) said: "The riding one should greet the walking one, the walking one should greet the sitting one, and the small number of persons should greet the large number of persons."
In the narration of Imam Al Bukhari, he said: "The young should greet the old, the passerby should greet the sitting one, and the small group of persons should greet the large group." [Sahih Al Bukhari No. 6232, 6234 and Sahih Muslim No. 2165].