Pikachu's trainer is Ash Ketchum and this is their story. On the morning of choosing a pokemon, Ash accidentally slept in. If Ash didn't sleep in then he would have probably gotten a different pokemon.He probably would have chosen Charmander, Bulbasuar or Squirtle but he missed out on all three of them and got Pikachu. Pikachu was the only pokemon left so Ash was forced to have him or not become a pokemon master until the following year.
Ash didn't want to wait to be a pokemon master so that is how he got Pikachu. Pikachu didn't like Ash to start of with, so he didn't listen to him and Ash didn't like Pikachu. At the start of their journey, they were attacked by a bunch of Spearow pokemon and to get away from them, Pikachu and Ash borrowed Misty's bike. They were badly hurt but Pikachu performed a thunder bolt and scared all the Spearow away. Ash then took Pikachu to the pokemon center and so Pikachu could recover and be on their way for the rest of their journey. They are now the best of friends.