Inculcate the habit of greeting each other by saying “As salamu ‘alaykum” and reply by saying “Wa ‘alaykumussalam”. All other methods of greeting are nonsensical.
The person who greets first receives more reward.
When a person conveys someone’s salam to you, reply by saying “‘Alayhim wa ‘alaykumus salam”.
If one person from a group of people makes salam, it will suffice on behalf of the rest. Similarly, if one person in a group replies, it will suffice on behalf of the rest.
Additional notes: It is prohibited to bow when greeting by hand. If you are at a distance and you greet someone or that person greets you, it will be permissible to use your hand. However, you should also make a verbal salam or reply. Muslim children who study in government schools should not greet in methods that are adopted by the west and non-Muslims. Instead, they should greet the teachers according to the Islamic method (i.e. if the teachers are Muslims). If the teacher is not a Muslim, one should merely say “salam” or “As salamu ‘ala manit taba’al huda”. The words “As salamu ‘alaykum” should not be used for the kuffar. These rules apply to all Muslims.