After leaving your hometown, Bianca follows to teach you how to catch Pokémon, specifically a Level 2 Purrloin (who, technically, is an illegal Pokémon, because it does not learn Growl until level 3). She demonstrates weakening the Pokémon first before throwing a Poké Ball for better results. She then leaves so that you can continue on your way.
Ledge Jumping
On a side note, you can jump over ledges to get to places faster. Just take notice that you can't go over ledges not facing your way.
With Floccesy Town in sight, Alder jumps from a ledge to talk to you for a minute. He asks that you come with him to a Trainer's School in said town and help with a few things, to which the player character agrees to. Good thing you are already on your way in anyway.
Patrat is a Normal-type Pokémon that is fairly common around the Unova region. Patrat isn't especially useful, although if you were to choose between it and Purrloin to catch at this very moment, it would be a better choice, as its moves work better earlier on and it can be a useful HM Slave, as it has access to Cut and its evolution Watchog can learn Strength. It can learn a few useful techniques early on, such as Detect, Crunch, and Hypnosis, although later on, better Normal-types, such as Clefable and Cinccino become available. Patrat can be fine for now, although eventually it'll become less useful.
Patrat evolves into Watchog at Level 20.
Purrloin is a Dark-type Pokémon that is pretty uncommon around the region, only being found around the earlier routes. Purrloin, like Patrat, overall isn't too useful to use for the whole game, although it does learn a couple of useful techniques early on, such as Sand-Attack and Assist. Like Patrat, there are better Pokémon of its type available later on, including Scrafty, Bisharp, Crawdaunt, and Krookodile. Purrloin and its evolution Liepard aren't really the best team members for this game, but if you want to give them a chance, go ahead.