Etiquettes of Drinking Zam-Zam Water
Ibn Abbas reported: I served. (water of) zamzam to Allah's Messenger (may peace be upon him), and he drank it while standing.
-Sahih Muslim [Book #023, Hadith #5023]
Ibn 'Abbas reported that Allah's Messenger (may peace be upon him) drank (water) from zamzam in a bucket while he was standing.
-Sahih Muslim [Book #023, Hadith #5024]
Ibn 'Abbas reported: I served (water from) zamzam to Allah's Messenger (may peace be upon him), and he drank while standing, and he asked for it while he was near the House (i. e. House of Allah-Ka'ba).
-Sahih Muslim [Book #023, Hadith #5026]
Facing the qiblah
Ibn 'Abbaas may Allaah be pleased with him said: "If you drink Zam-Zam water drink your fill, turn towards the Qiblah, say "Bismillaah" (in the name of Allaah) and breath three times and when you finish, you should say "Alhamdu lillaah" (All perfect praise be to Allaah)."
The Sunnah when drinking Zam-Zam or any other drink is to drink by three sips.
Imaam Al-Bukhaari and Muslim may Allaah have mercy upon them reported that the Prophet sallallaahu alayhi wa sallam ( may Allaah exalt his mention ) used to drink in three sips, breathing outside the container. Ibn Qataadah may Allaah have mercy upon him also narrated that the Prophet sallallaahu `alayhi wa sallam ( may Allaah exalt his mention ) forbade breathing into the container when drinking. When a person drinks in one go, he will possibly breath into the inside of the container (the thing which is forbidden).(Source)
A person drinking Zamzam water should intend and hope for healing, blessings and whatever is best for him in this life and in the hereafter.
The Prophet (saws) said, "The water of Zam-Zam is (good) for whatever it is intended."Source
Supplicate in between drinking
Make as many supplications as one wishes between the gulps. The Prophet(PBOH) taught a specific dua as follows
اللَّهُمَّ إِنِّي أَسْأَلُكَ عِلْمَاً نَافِعَاًً وَرِزْقَاً وَاسِعَاًَ وَشِفَاءً مِنْ كُلِّ دَاءٍ
"O Allah, I seek beneficial knowledge, wide sustenance and cure from all ailments from You".
Say Alhamdulillah after drinking
But, there is a controversy between scholars about standing during drinking zam zam water because of this reason.