There are a few Story Breadcrumbs in this area that I've never seen people mention for some reason.
For one thing, Red's motivation for leaving town is because he peeked at Prof. Oak's computer and read his E-mail.
It contained a message from the Indigo Plateau, saying that the Elite Four are accepting challenges again, (and by the way, they want the Professor to drop by).
Being 10, and not finding the Professor anywhere in town, Red decides that he must have left already and starts leaving town-
By the way, the 5 pokeballs were given to you by Prof. Oak's Aide when you tried to leave the lab again, in an event added in the remakes.