What are the major events that occur after the arrival of the trains at Birkenau and before the evacuation of Buna?

Their arrival at Birkenau marks the beginning of Elie's and his father's nightmarish experience at four different Nazi concentration camps. Birkenau acted as a "reception center" for the larger Auschwitz camp. The first major event at Birkenau was the selection. Directed by the infamous Dr. Mengele, Jews were pointed in two different directions, either to their immediate death in the gas chambers or to slow death as workers for the Third Reich. Luckily, just before the selection, Elie is told by another prisoner to lie about his and his father's age. Instead of saying 14, Elie tells the Nazis that he is 18 and that his father (actually 50) is 40. They are pointed to the left, and for a short time, they believe they are being sent to their deaths as they witness the bodies of dead children being tossed into fiery pits. Just short of the pits, they are turned toward a series of barracks. As he enters, Elie proclaims his loss of faith in God:

Never shall I forget those flames which consumed my faith forever. . . . Never shall I forget those moments which murdered my God and my soul and turned my dreams to dust.

Once they reach the barracks, the Jews are told to shed their clothes except for their shoes and their belts. Barbers shave the men, and they are given prison clothes, often of the wrong size. All the while they are being beaten with truncheons by the "Kapos," prisoners who have been chosen to keep discipline. During the night, they are marched a short distance to Auschwitz where signs read "Work is Liberty." They stay at Auschwitz for three weeks with very little to do. Elie and his father encounter a relative who is worried about his family. Elie lies to the man, telling him his family is safe.
At Buna, Elie and his father go to work sorting electrical parts. Elie has a conflict with another prisoner about his shoes, which he ultimately loses in an effort to protect his father from beatings. They live with a group of musicians, including the violinist Juliek who will later die on the forced march to Buchenwald. Another prisoner, Franek, notices Elie's gold crown and demands it. When Elie makes excuses, Franek brutalizes his father, and Elie is ultimately forced to give up the crown, extracted from his mouth with a rusty spoon in the lavatory. Similarly, when Elie accidentally discovers the Kapo Idek having sex with a young Polish girl, he is whipped.
Signs of the impending Nazi defeat are signaled when bombers appear overhead and several of the camp's buildings are destroyed. It is also at this time that a "black gallows" appears in the camp. When a young boy is hanged in front of the prisoners, a man asks, "Where is God now?" to which Elie silently answers, "Where is He? Here he is—He is hanging here on this gallows."
Elie again demonstrates his condemnation of God during the Jewish holidays Rosh Hashanah and Yom Kippur when he does not fast. Not long after, the prisoners are again subjected to the dreaded selection. Elie worries about his father, who has become quite weak during their ordeal. There is a poignant scene in which Elie's father gives his son his knife and spoon as a way of saying goodbye. Miraculously, however, Elie's father escapes the selection. He is in the barracks when Elie returns from work.
In his final days at Buna, Elie's foot becomes badly swollen, and he undergoes surgery. While in the hospital, Elie learns that Russian troops are advancing, and the entire camp will be evacuated. Elie and his father then face a difficult decision: either stay in the camp's infirmary or be evacuated with the rest of the prisoners. They decide to evacuate. Unfortunately, they choose poorly because the camp is ultimately liberated by the Russians. All those who stayed in the camp's hospital were freed. Elie and his father then begin the terrible forced march to Buchenwald.