The first album I picked out for myself was the Ghostbusters soundtrack on LP. I guess I was around 7 or 8. I don't remember if it was a reward for something or my mom just being nice and letting me get it. I do remember picking it out because the LP sleeve was just a big Ghostbusters logo.
Around 9 or 10 years old I was getting an allowance and bought a cassette of Dj Jazzy Jeff and the Fresh Prince : He's the DJ, I'm the Rapper (I had to look it up based on the songs I bought it for - Nightmare on My Street (I was obsessed with any thing Freddy Krueger) and Parents Just don't Understand).
By 12ish I was into punk and not really sure what came first but I remember getting a Fugazi and Misfits album and wearing them out.
By the time I was 16 and got a real part time job I started buying roughly an album per week. My tastes ranged from the grunge Sub Pop stuff, to classics like Led Zeppelin, to Dr.Dre. I really don't remember what the first purchase of those years was.