Praise be to Allah.
We appreciate your eagerness to learn; may Allaah increase us and you in knowledge and acceptable righteous deeds. It was reported from Muaawiyah ibn Abi Sufyaan (may Allaah be pleased with them both) that the Prophet (peace and blessings of Allaah be upon him) said: Whomever Allaah wills good for, He enables him to understand the religion.(Agreed upon).
In response to your question: His Excellency Shaykh Abd al-Azeez ibn Baaz (may Allaah have mercy on him) was asked the following question:
Nowadays, as a result of dealing with the West and the East most of whose people are kuffaar of different sects and religions they say the greeting of Islam (as-salaamu alaykum) when they meet us in any place. What should we do with regard to them?
He (may Allaah have mercy on him) replied:
It was reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: Do not initiate the greeting of salaam to the Jews and Christians, and if you encounter them in the street, push them to the narrowest part of the road.(Narrated by Muslim). And he (peace and blessings of Allaah be upon him) said: If the people of the Book greet you with salaam, say Wa alaykum (and also upon you). (Agreed upon). The People of the Book are the Jews and Christians; the ruling concerning the rest of the kuffaar is the same as the ruling concerning the Jews and Christians with regard to this matter, because there is no evidence (daleel) to indicate anything different.
So we should not initiate the greeting of salaam to a kaafir at all; if he initiates the greeting of salaam, then we should say wa alaykum in response, in obedience to the command of the Messenger (peace and blessings of Allaah be upon him). There is no reason why we cannot then say, How are you? Or, How are your children? as has been permitted by some of the scholars, including Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him), especially when doing so serves an Islamic purpose such as encouraging him to take an interest in Islam so that he will be receptive to dawah, because Allaah says (interpretation of the meanings):
Invite (mankind, O Muhammad) to the way of your Lord (i.e. Islam) with wisdom (i.e. with the Divine Revelation and the Quraan) and fair preaching, and argue with them in a way that is better [al-Nahl 16:125]
And argue not with the people of the Scripture (Jews and Christians), unless it be in (a way) that is better (with good words and in good manner, inviting them to Islamic Monotheism with His Verses), except with such of them as do wrong [al-Ankaboot 29:46]
(Majmoo Fataawaa wa Rasaail Samaahat al-Shaykh Abd al-Azeez ibn Baaz Fataawaa al-Aqeedah, part 2, p. 1042)
Imaam Ibn al-Qayyim (may Allaah have mercy on him) said, with regard to initiating the greeting with a kaafir: A group i.e., of scholars said: it is permissible to initiate the greeting for a purpose, such as some interest that one hopes to achieve, or because one fears his harm, or because they are related by blood, or for some other reason that dictates that.