Anas (May Allah be pleased with him) reported that he passed by some children and greeted them. Then he said: “Messenger of Allah (sal Allahu alayhi wa sallam) used to do the same.” [Al-Bukhari and Muslim]
The sunnah is to give the salaam to everyone you meet, even if they are younger than you. The Prophet (sal Allahu alayhi wa sallam) would begin with the salaam whenever he met someone while he was the most important person among the people.
So when you initiate the salaam to those you meet, you are gaining a lot of good from it, such as:
1- following the Prophet sal Allahu alayhi wa sallam.
2- this will be a way for you to spread a sunnah that has diminished greatly among Muslims. It is known that the reviving of a sunnah that has been forgotten receives two rewards - one for doing the deed, and one for reviving the sunnah.
3- You will be the reason for the response of the person to your greeting, and this response to the salaam is fardh (kifayah), so you will be the cause of this person completing an obligatory act.
So for these reasons the initiating of the salaam is greater than its response, even if the response is a fardh, because the fardh is dependent on the sunnah, so doing the sunnah is better than the fardh.
This is a topic that may cause some confusion as it has among some scholars because it is agreed upon than the fardh(obligatory) is better than the sunnah - for example, the two raka’ahs of salatul fajr are better than the two sunnah ra’akahs before it because it is fardh while the other two are sunnah. In this case however, initiating the salaam is a sunnah and is better than responding to it because its response is dependent on its initiation.
-Shaykh ibn Uthaymeen (rahimahullah) Sharh Riyad as-Saliheen.