[If it is not explicitly spelled out otherwise, assume all events that are handwaved to be near exact parallels to canon]
It is said that the phoenix is immortal, as it is immediately reborn once it dies. This is not always the case. To reproduce, a phoenix will burn its body and use its ashes to warm an egg. This was the case with Yellow, born a phoenix in the Viridian Forest. "Uncle" Wilton encountered the fledgling in the nest of ashes, and took pity on the helpless bird. The people of Viridian were comfortable with the creatures in the forest - the creatures were peaceful or so small that they posed no threat. As a result, even when Wilton found that Yellow was a fledgling phoenix, he did not kill her nor attempt to sell her for a profit. He raised her until she could fend for herself, then left her be. She later rescued him from pokemon defending their homes during one of his fishing trips, creating a lasting friendship between the two.
Because of Wilton's relationship with the phoenix, she began to understand human customs. She began to harness enough magic to manifest as a human child, in order to communicate properly with Wilton, who taught her how to fish and how to speak. She lived peacefully in the forest, guiding lost travelers through it when they needed her most.
One such traveler was Red, during his first days of following Professor Oak's mission. Though he was disguised as a human, she immediately saw that he was something different entirely, and was eager to guide him to safety. (Green, on the other hand, had her hiding. Vampires are sort of... dangerous, after all.) She didn't see either of them again for quite some time. Red was the first encounter she had after the initial meeting - he had returned to Viridian to challenge the gym leader, going through the forest to return. The forest had since become very dangerous, with the pokemon and some of the lesser magical creatures starting to go mad and bloodthirsty. She was rescued from the forest by Red, and stayed in town under her human guise to avoid suspicion. After the defeat of Giovanni, she found and brought Red to the little town clinic, offering her assistance in the form of her natural blessing - a healing tear. He regained his strength and resumed his mission.
It was not until Red disappeared that Yellow was called upon. She met Blue, a changeling (a thieving sort - it was just lucky on Yellow's part that she had nothing of value to steal), who explained to her about the world - about how Viridian City's generally peaceful relations with supernaturals was not commonplace, and about how there were some supernatural creatures that certainly weren't helping. The Elite Four had issued a challenge to Red, and he had since vanished. Fearing that he was dead or injured beyond assistance, Blue sought Yellow's ability to heal and possibly rebirth, and Yellow agreed because of the admiration and general fondness she held for the unicorn. Gathering her courage and asking pokemon from her forest to accompany her, she wore the disguise that Blue gave to her, taking with her a rattata.
She encountered several enemies, but it became very quickly obvious that her meager experience in fighting (which was comprised mostly of evasion tactics) was not going to be suited to her assigned mission. It was then that she was reunited with Green, who explained to her that she needed to learn how to fight in order to save Red. Though hesitant, she agreed to learn from Green, and trained with him as long as she could. She realized that he was not as frightening as he had appeared - not quite the horrific killer he appeared, simply a being that killed out of necessity. Because of his generally kindhearted (if tsun) nature, she quickly grew fond of him and began to respect him all the same.
Lance, a dragon wearing a human face in order to inform the beasts of their imminent demise, stole the dragonair that was to sail her over the seas. A brief battle with Lance showed Yellow that she was significantly weaker, not simply in terms of experience. Her strengths were dampened and her abilities smothered in his presence. She was a creature of rebirth, of giving life... Lance was a being of death, be it dealt through chaos or through "justice". He escaped with the dragonair, leaving Yellow to resume her journey with Pika, receiving only Pika's new ability to surf in consolation.
Making her way to Cerise island, she watched as her friends battled against the members of the Elite Four. She herself then challenged Lance, once they were mostly dealt with, and began her own battle against him. Giovanni tried to help, but failed. Red tried to jump in, but was hideously burned by Lance's flames. Yellow attempted to fight alongside her pokemon, but her left arm broke as a result of getting tangled in Lance's claws. It wasn't until Lance summoned Lugia that Yellow discovered and unleashed her true natural strength - bestowing her strength upon her pokemon. She channelled all of her magic into Pika, and with similar contributions from Red, Blue, and Green's pokemon, Pika managed to accomplished the powerful MegaVolt, striking Lance and breaking his hold on the mighty legendary pokemon.
After Lugia's retreat, Yellow tended to everyone's wounds. All wounds... save for Red's. He had sustained burns from Lance that Yellow could not heal - Lance's powers, even after his defeat, were beyond Yellow's ability to overcome. Red travels into Johto's mountains to find a cure. Finding nothing much else to do, she goes to visit Wilton for the first time outside of the forest, and he excitedly greets her. He tells her about the sights to see and the legends to investigate, and she happily goes through Johto to explore them.
She resumed her journey as per the established canon. Differences in the GSC arc will likely be found in the histories of the Johto Dex Holders.