Hard question.
I think the first console I bought with my own money was the GameCube. I was 15 at the time, bought it shortly after the price cut to €99. However, it could also be one of my GameBoy models. I know my classic model was a gift, but I could have bought either the GameBoy Pocket or the GameBoy Color myself. I got those in the span of 1997-1999, when I was 9 to 11. I'm leaning towards GameCube being my first true buy though.
First game I bought myself is even harder. I just don't know, but let me think out loud here. I got my GameCube with Zelda: The Wind Waker and either Mario Sunshine or Luigi's Mansion, so naturally I bought those myself too. However, I'm pretty sure I've been buying games myself since quite a while before that. I bought SimCity 4 myself earlier that year, I remember pre-ordering it (a first for me), and Warcraft III that same year. I know the first game I was truly ever 'hyped' for (so, keeping up with development and anticipating it's exact release) was Zelda: Ocarina of Time on the N64. However, that game was still a gift, so it's probably a game that falls in-between.
In those days most of what I got was still a gift, even if I knew what I wanted I just had to ask for it. Though I'm pretty sure I got Pokémon Stadium and Zelda: Majora's Mask myself. Of those, Pokémon Stadium is older, so for now I'll go with that.
Long story short; GameCube and Pokémon Stadium (N64).
EDIT: Although... Did I get Pokémon Blue or Link's Awakening by myself? Pokémon maybe, maybe that's the first actually... Mmm...
