What to Say when Giving Salam
Question

I want to ask about how to give and return the greeting of Salam, as it was narrated from the Prophet Muhammad (blessings and peace of Allah be upon him). Was the response “Wa `alaykum As-salam wa rahmatullahi wa barakatuhu wa maghfiratuhu (and upon you be peace and the mercy of Allah and His blessings and His forgiveness)'' narrated from him?
Praise be to Allah.What to say when giving and returning Salam 
The Muslim may limit the greeting when initiating it to simply saying “As-salamu `alaykum (peace be upon you).” But if he adds “wa rahmatullah (and the mercy of Allah),” that is better. And if he also adds to that “wa barakatuhu (and His blessings),” that is even better.
The one who is greeted may limit his response to the greeting by saying the same thing with which he was greeted. However, if he adds more to that, it is better, because Allah, may He be exalted, says (interpretation of the meaning):
{And when you are greeted with a greeting, greet [in return] with one better than it or [at least] return it [in a like manner.]} [An-Nisa’ 4:86]
Ibn `Abbas (may Allah be pleased with him) narrated that `Umar (may Allah be pleased with him) came to the Prophet (blessings and peace of Allah be upon him) when he was in an upper room of his, and he said: As-salamu `alayka ya Rasul-Allah, As-salamu `alaykum, may `Umar enter? (Narrated by Abu Dawud (5203); classed as authentic by Al-Albani in Sahih Abu Dawud)
At-Tirmidhi (2721) narrated that the Prophet (blessings and peace of Allah be upon him) said: “When a man meets his Muslim brother, let him say As-salamu `alaykum wa ramatullahi wa barakatuhu (peace be upon you and the mercy of Allah and His blessings).” This Hadith was classed as authentic by Al-Albani in Sahih At-Tirmidhi.
`Imran ibn Al-Husayn (may Allah be pleased with him) said: A man came to the Prophet (blessings and peace of Allah be upon him) and said: As-salamu `alaykum. He (blessings and peace of Allah be upon him) returned the greeting, then he (the man) sat down, and the Prophet (blessings and peace of Allah be upon him) said: “Ten [meaning ten rewards for a good deed].” Another man came and said: As-salamu `alaykum wa rahmatullah. He returned the greeting , then he (the man) sat down. The Prophet (blessings and peace of Allah be upon him) said: “Twenty.” Another man came and said: As-salamu `alaykum wa rahmatullahi wa barakatuhu. He (blessings and peace of Allah be upon him) returned the greeting, then he (the man) sat down. The Prophet (blessings and peace of Allah be upon him) said: “Thirty.” (Narrated by Abu Dawud (5195) and by At-Tirmidhi (2689), who said: It is sound. It was classed as authentic by Al-Albani in Sahih At-Tirmidhi.)
`Aishah (may Allah be pleased with her) said: The Messenger of Allah (blessings and peace of Allah be upon him) said to me: “Jibril conveys greetings of Salam to you.” I said: And upon him be peace and the mercy of Allah. (Narrated by Al-Bukhari, 3045; and Muslim, 2447)
An-Nawawi (may Allah have mercy on him) said in the chapter on how to give the greeting of Salam:
“It is recommended for the one who initiates the greeting to say: “As-salamu `alaykum wa rahmatullahi wa barakatuhu,” using the plural form of the pronoun (-kum, meaning you), even if the recipient of the greeting is just one person.
The one who responds should say: “Wa `alaykum As-salam wa rahmatullahi wa barakatuhu,” with the conjunction (wa, meaning and) in the phrase “wa `alaykum (and upon you).” (Riyadh As-Salihin  p. 446)
Ruling on adding ‘wa maghfiratuhu’ when returning Salam
With regard to adding “wa maghfiratuhu (and His forgiveness),” it is mentioned in some Hadiths with regard to initiating the greeting and returning it, but these reports are not sound. Among the Hadiths which mentioned that are the following:

It was narrated from Sahl ibn Mu`adh ibn Anas (may Allah have mercy on him), from his father, from the Prophet (blessings and peace of Allah be upon him)… A Hadith similar to the Hadith of `Imran quoted above, with the addition: A fourth man entered and said: “As-salamu `alaykum wa rahmatullahi wa barakatuhu wa maghfiratuhu.” The Prophet (blessings and peace of Allah be upon him) said: “Forty.” Then he said: “Thus are good deeds rewarded.” (Narrated by Abu Dawud (5196). This Hadith and the addition “wa maghfiratuhu (and His forgiveness)” was classed as inauthentic by Ibn Al-`Arabi Al-Maliki, An-Nawawi, Ibn Al-Qayyim, Ibn Hajar and Al-Albani (may Allah have mercy on them))

Ibn Al-Qayyim (may Allah have mercy on him) said:
This Hadith is inauthentic, because it has three defects:

It is narrated by Abu Marhum `Abd Ar-Rahim ibn Maymun, whose Hadith cannot be quoted as evidence.
Its Isnad (chain of narrators) also includes Sahl ibn Mu`adh, to whom the same comment applies.
Sa`id ibn Abu Maryam, one of the narrators of the Hadith, was uncertain in his narration. In fact he said: I think I heard Nafi` ibn Yazid... (Zad Al-Ma`ad fi Hadiy Khayr Al-`Ibad, 2/417, 418; see: As-Silsilah Ad-Da`ifah, 5433)


Anas (may Allah be pleased with him) said: A man would pass by the Prophet (blessings and peace of Allah be upon him) and say: “As-salamu `alayka ya Rasulullah (peace be upon you, O Messenger of Allah),” and the Prophet (blessings and peace of Allah be upon him) would say to him: “Wa `alayka As-salam wa rahmatullahi wa barakatuhu wa maghfiratuhu wa ridwanuhu (and upon you be peace and the mercy of Allah and His blessings and His forgiveness and His pleasure).” It was said to him: O Messenger of Allah, you greet this man with a Salam with which you do not greet any of your companions. He said: “Why shouldn’t I do that, when he leaves with the reward of umpteen men?” And he used to take care of his companions. (Narrated by Ibn As-Sunni in `Amal Al-Yawm wa’l-Laylah (235). It is a very inauthentic Hadith; it was classed as inauthentic by Ibn Al-Qayyim in Zad Al-Ma`ad (2/418) and by Ibn Hajar when he said: “Ibn As-Sunni narrated in his book with a flimsy Isnad (chain of narrators) from Anas, who said: A man would pass by…” (Fath Al-Bari, 11/6)
It was narrated that Zayd ibn Arqam (may Allah be pleased with him) said: When the Prophet (blessings and peace of Allah be upon him) greeted us, we would respond: “Wa `alayka As-salam wa rahmatullahi wa barakatuhu wa maghfiratuhu.” (Narrated by Al-Bayhaqi in Shu`ab Al-Iman, 6/456) He classed it as inauthentic by saying: “If this was sound, we would accept it, but in its Isnad going back to Shu`bah there are some narrators whose Hadith is not to be quoted as evidence.”

Ibn Hajar (may Allah have mercy on him) said:
“Al-Bayhaqi narrated in Shu`ab Al-Iman with an inauthentic Isnad from Zayd ibn Arqam… And he mentioned the same Hadith." (Fath Al-Bari,  11/6)
Based on that, the best and most complete phrase to use when giving the greeting of Salam is: “As-salamu `alaykum wa rahmatullahi wa barakatuhu” and the best phrase to use when returning the greeting is: “Wa `alaykum As-salam wa rahmatullahi wa barakatuhu.”
For more details, please see this category: Manners of Greeting with Salam
And Allah knows best.