The first game that I owned was Pokemon Stadium which I received as a birthday present in May 2000. Following that, I finally got Red and Blue at Christmas 2000 which I used to play through Stadium through use of the Transfer Pak because I didn't get a Game Boy Color until my following birthday in 2001 along with Pokemon Silver.
That said, I had been watching over the shoulders of fellow school and class peers with great interest while they played Red and Blue on their GB Pocket/Color/Classic at Primary School as early as when the Pokemon craze went mainstream in the UK in 1999. I was also hooked on the anime as soon as it hit UK terrestrial television in late 1999, therefore my love for Pokemon began many months prior to me obtaining my first Pokemon game.
I started playing Blue back in 1999 towards the end of my seventh grade year. My little bro got Red, and then that Christmas we both got the Pikachu GBC bundle with Yellow. Since then I've owned at least one game per generation.