Regardless of which gender is chosen and how old the Vault Dweller is chosen to be on a scale of 16 to 35, the Vault Dweller is canonically male as shown in the Book of the Elders, and is implied to canonically be Albert Cole due to Interplay's concept-art for the Vault Dweller and his appearance in Fallout: Brotherhood of Steel, suggesting that the Vault Dweller is 27 years old at the beginning of Fallout 1.
Believing that someone must be artificially creating the super mutants and fearing for the safety of the Vault, he ordered the Vault Dweller to find and destroy the source of the mutants. The Vault Dweller soon found an old military base where humans were being dunked into vats of Forced Evolutionary Virus in order to be converted into super mutants. He destroyed both this base and the Master of the super mutants beneath the Cathedral in Boneyard before returning to the Vault, only to be told that he saved the lives of the vault and maybe even the human race but is forced out because of Jacoren's fear that everyone in the vault would want to leave.
After leaving the Vault for a final time, the Vault Dweller wandered the desert, but never moved far from the mountains that shielded the Vault from the rest of the world. Some other members of the Vault, upon hearing of what happened, decided to leave the Vault and join the Vault Dweller, following him soon afterwards. The Vault Dweller eventually found and rejoined this small group of former vault dwellers. Knowing little of the outside world, the group would almost certainly have died had it not been for the Vault Dweller's assistance. At this time, the Vault Dweller removed his Vault suit and had it placed in the Temple of Trials and never wore it again.
Two months later, the Vault Dweller headed north, to the great canyons, with a small group of vault dwellers and wastelanders and founded a small village. Initially, they would send scouts back towards the Vault, to help others who thought like themselves, but that slowly came to an end. The Vault Dweller taught the others the skills they would need to survive and grow strong. Hunting, farming and other skills to feed them, engineering and science to build their homes, fighting to protect what was theirs. Slowly, they took the knowledge that the Vault Dweller had learned on his adventure and together, they learned to thrive. Over time, their rag-tag group had turned into a tribe. The Vault Dweller fell in love with one of them, named Pat, and they raised a family, like all of their tribespeople. The construction of Arroyo was completed on August 18, 2167.
The Vault Dweller and Pat led the tribe which grew larger and stronger. In 2188, their daughter, who would later become the Village Elder, was born. Eventually, she and the other children became the leaders of the community.
On January 16, 2208, several years after Pat's death, the Vault Dweller wrote his memoirs. Soon after that, he vanished from Arroyo and was presumed dead. The Vault Dweller left the Vault suit behind, folded on the bed. Some say the Vault Dweller was taken by the sky spirits, others say that the Vault Dweller felt it was time to move on and leave the Elders to guide Arroyo to its destiny. After the One-Moon (Month) Cycle of mourning for the Vault Dweller ended, activity in Arroyo began to return to normal. Years later, the Vault Dweller's grandchild, later called the Chosen One, was born.
After leaving Arroyo the only known place where he was spotted was the town of Carbon, where he is known simply as the Wasteland Stranger. There, he helped some of the people to hide from raiders and also helped the Initiate trace the remnants of the Master's army and some Brotherhood of Steel paladins, directing him to Los. We don't know what became of him afterwards, but it is certain he left Carbon and continued his travel in the Wasteland.
Interactions with the Wasteland Stranger during his stay in Carbon can result in the Initiate receiving from him a Red Ryder BB gun. Additionally, the Initiate may find a Vault 13 flask in the Carbon Mill, at which the Wasteland Stranger remarks at being a "sentimental old fool" and offers a trade for it.
Besides Arroyo and the Chosen One, the Vault Dweller's journey lived on to the extent that he is still revered by a fair portion of the people in the Wasteland. His statue is prominently displayed in the New California Republic capital, Shady Sands.
He is remembered under several names, including "the Vault Dweller" and "the Stranger" (used by those who - while believing in his greatness - doubt he ever could have come from a Vault). He refers to himself as "the Wanderer" at the end of his memoirs. Children who heard of his story would often play at finding Vault 13, some of these children even grew into adults obsessed with proving the Vault Dweller's former home existed. For a time, President Tandi would even sponsor such adventurers, until giving up after many years of failure.
I always thought of Fallout's story of drawing straws was something the Overseer thought of. "Look, someone has to go outside, and they may die out there, so let's draw straws". Someone, probably not the Overseer since he wouldn't have gone outside no matter what, demanded that EVERYONE draw a straw in order to be truly fair. But to be honest, the vault dwellers were simply terrified of going outside, and terrified people do not always make the wisest decisions.
Anyway, when big dumb Larry drew the straw, maybe the Overseer saw a way to rid their limited gene pool of some really bad alleles.