The start of the game could begin with you graduating from school, and after waking up from graduation from Pokemon School, your Mum could congratulate you with a Pokemon, you would be able to choose of course. So you go to show your rival/friend or whatever, and he/she exclaims how he/she wished she had a Pokemon, they were top of the class and he/she was one of the only students that doesn't have one. The Professor could overhear while walking past, and would be surprised that their top student didn't have one and decides to give them one as a gift.
Then,you and all your classmates head off on route one together, to start your epic journey. That would be awesome.