The Prophet (peace and blessings be upon him) said, “Islam is built upon five pillars: testifying that there is no true god except Allah and that Muhammad is the Messenger of Allah, performing Prayer, paying the zakah, making the pilgrimage to the Sacred House (Hajj), and fasting the month of Ramadan” (Al-Bukhari). Also, we’d like to commend your pursuit of Islamic counseling.
The minimum number of days required to perform the Hajj is three. Elaborating on this, we cite for you the following:
The prescribed acts of pilgrimage or Hajj are actually divided into pillars (arkan) and obligations (wajibat). If any of the pillars is missed for one reason or another, then the whole pilgrimage is missed and the sacrifice of a lamb does not substitute or qualify to redeem the missed Hajj. As to the obligations, if any of them is missed, it has to be substituted with the sacrifice of a lamb.
For instance, standing at `Arafah is one of the basic pillars for pilgrimage and if a person does not stand at `Arafah, his or her pilgrimage is annulled. On the other hand, staying at Muzdalifah is an obligation; if missed, it can be redeemed with the sacrifice of a lamb. In this respect, pilgrimage regulations are numerous and Muslim scholars differ among themselves as to what is an obligation and what is a sunnah. Still, the shortest time to perform Hajj is for a person to stand in `Arafat, starting between the Zhuhr Prayer on Dhul-Hijjah 9 until Fajr of the tenth. So even if a person stands at `Arafat one minute before Fajr, then the pilgrimage is performed. As to the other rituals, such as the Tawaf and Sa`i (going around the Ka`bah and walking between Safa and Marwa), they can be performed in about one hour. Then the pilgrim makes tahallul (putting off ihram attire), and his or her Hajj is complete.
Regarding the other obligations, such as throwing stones during the three days of `Eid and staying at Mina, they are considered by some scholars to be obligations and by others to be sunnahs.
Allah Almighty knows best.