First game I bought myself with Christmas or birthday money I think was Super Mario RPG on SNES. I'm not entirely sure though, may have also been an N64 or PS1 game(Goldeneye and Legend of Legaia come to mind).
First I bought with my own money from a real job I believe was Yoshi's Island: Super Mario Advance 3 for GBA. I'm about 95% sure on this one, I had to wait 30-45 mins for my ride to pick me up every day and got it to play while waiting.