Yes, in Pokemon Yellow your Pikachu follows you around. It refuses to enter the pokeball, just like the cartoon which is the basis of the game. Yellow is a modified version of Red/Blue with Pikachu as the starter and an additional building. The small house at the northern beach of Route 19 that wasn't in Red/Blue is home to the "Surfing Pikachu" mini-game. Most of the features in Red/Blue/Yellow is tied to Pokemon Stadium for N64. Completing certain achievements there would grant you special Pokemon not retrievable any other way.
Ive got both Pokemon White 1 and white 2 And i also have Keldeo and Genesect which i gave to a friend to help train him.