Running Shoes are a held item in the game, and they are not obtained by catching Pokémon.
Here's how you get Running Shoes:
1. Talk to the Old Man in Viridian City: After beating Brock at the Pewter Gym, an Old Man will be standing in front of the Pokémon Center in Viridian City.
2. He'll give you the Running Shoes: Talk to him, and he'll give you the Running Shoes as a gift.
With Running Shoes, you'll be able to run faster throughout the game!