It seems that your question is referring to when Elie and his family are all still together in Sighet rather then when Elie and his father have already been through their camp experience. If that is the case, the Wiesel family and the other Jews from Sighet miss numerous opportunities to escape before their deportations. Below are some of those opportunities.
1. Moshe the Beadle narrowly escapes death when he is deported as a "foreign" Jew and returns to Sighet to warn everyone of the Nazis' intention to kill Jews. No one takes Moshe seriously; if the Wiesels had listened to him, they would have had plenty of time to leave before the Germans came.
2. Chlomo Wiesel, Elie's father, received word that he could gather his money and buy a safe escape for him and his family. Unfortunately, he does not think events are bad enough to do so.
3. When steps of dehumanization begin in Sighet such as wearing the star, quartering soldiers, moving to the ghetto, being forbidden to attend the synagogue, Elie's father does not recognize those steps for what they are. In Chapter One, he brushes them off by saying that "it's just a yellow star."
4. Even on the night before their deportation, a friend knocks on the Wiesels' window to try to get them to escape to safety, but they do not leave because they do not think that deportation to a "work" camp will be that bad.
Towards the end of the war it is clear that the camps are getting ready for evacuation as the Russian troops will be reaching the camp soon. This initiates a very difficult decision that must be made by Elie and his father. They must choose to go with the rest of the prisoners who are being evacuated, or they must choose to stay in the hospital, where thanks to Elie's wounded foot, he will be able to stay with his father:
The choice was in our hands. For once. We could decide our fate for ourselves. To stay, both of us, in the infirmary, where, thanks to my doctor, he could enter as either a patient or a medic.
Having heard rumours that all those staying would be killed or that the Germans would have mined the camp to explode it when they left, Elie convinces his father to go. However, it is only after the war that he discovers that those that stayed were liberated two days after the evacuation.