After obtaining the Helix Fossil, the Hive Mind continued to navigate RED through Mt. Moon. After a couple more battles, RED exited the dreaded Mountain, and began heading towards Cerulean City.
It's at this point in the play-through where RED started selecting the Helix Fossil during battle.
RED catches a wild Rattata and nicknames it JLVWNNOOOO, the Hive Mind almost immediately starts calling it "Jay Leno", a reference to the popular late night show host of the same name. RED also catches a Spearow and a Drowzee.
RED enters Cerulean City and heads towards the local Gym. RED challenges the Gym Leader, Misty, and promptly loses. After 2 hours of little to no success, RED finally defeats Misty and is awarded the Cascade Badge.
RED exits the Gym and heads north towards the Nugget Bridge.
After nearly 3 1/2 hours of defeating trainers, Pidgey evolves into Pidgeotto. The Hive Mind and RED manage to reach Bill's house, and receive the S.S. TICKET.
With 5 minutes if the day remaining, RED catches a wild Rattata. The Rattata would later be known as Dig Rat.