Gen 1: You have to go on an errand after attaining your First Pokemon from Professor Oak as customary in that town/region. After the Errand Oak encourages you to test your skills as a trainer and to seek fame and Glory. TR story doesn't start until Mount Moon. (After First Gym)
Gen 2: You are running an Errand for Elm and given a Pokemon to defend yourself from wild Pokemon. After you finish your Errand you are (again) encouraged to take the League Challenge. You don't meet Team Rocket for the main story until Azalea. (After First Gym)
Gen 3: You hear Birch's cries for help, after getting a Pokemon ans showing battle skills he asks you to assist him and his daughter with their research, later (after talking to your dad another city over) you are encouraged to battle the league. You meet the main villan and start the main story after beating Roxanne and assisint the Devon Corp guy and Mr. Briney, saving them from Mascot team. (After First Gym)
Gen 4: You and your close friend seek adventure. (In D/P you leave to search for the Legendary Lake Pokemon after watching the news report on the Red Gyarados. In Pt you leave to ask Prof. Rowan questions. You are given your Pokemon (or retrieved them at the lake) and are asked to come by Rowan's lab later. Once there you are encouraged (AGAIN) by others to take part in the Pokemon League. You meet the main villans and get started on the main story after returning to Jubilife (After the first Gym battle) and assisting Rowan's assistant in a battle against Galactic Grunts. (Pt you meet Cyrus at the lake. which changes part of it though you do not know him as the main bad guy yet. First real encounter with Galactic is in Jubilife)
Gen 5: You receive Pokemon from the Professor as a Birthday gift and share with your friends. After going to thank the prof. for the gifts she asks you to assist her, and later you and your friend start to walk down the path of the League Trainer. You meet Plasma in the next town (breaking the cycle even though first battle encounter isn't until Dream Yard after the First Gym).
Following the cycle, I'm gunna ball park, You and your rival receive your Pokemon while attending the Trainer school. After receiving starter you are challenged by your rival to challenge the Gyms and leave on your journey.
The new Professor can recruit the two characters to assist with his study of Pokemon "Strengths".
I don't see it being too far off from this scenario. Maybe different reason for leaving, but I doubt its to save the world as that always comes from incidental build up. Reoccurring villains that just build up your will to defy them and stop their evil deeds before they harm everyone.
Not to mention the local police and people of power are just going to force you to do it anyway like they always do.