debated a few times about whether to answer this question, because i've never been as interested in music as people are, or at least i'm not interested in music in the same way people tend to be.
i think, depending on how we define it, the first album i ever bought myself was either:
1) the two-record version of

the original cast recording of CATS.
finding it at all was pure chance, i hadn't sought it out. i bought it immediately without question, and didn't even own a record player.
fun fact: the broadway recording (which is on these records) contains growltiger's last stand, a song that was not performed for the
vhs film (which, yes, i also own and have watched many a time) and i'm glad i got to have some recording of growltiger because it was a good song about an interesting cat!
the vhs version was cut down quite a bit, i think for length (misoffelees, and mcavity lose at least a verse each, and the jellicle ball is shortened significantly), but growltiger being entirely gone from it is a shame. i think it was chosen to cut over, say, jennyanydots because growtilger doesn't fit easily into any sort of story you could tell with the songs and cats on offer, and he stands out quite a bit, and leaves more room to position mcavity as a villain. but i am sad about that decision, because i really would have liked to see it performed by the vhs cast. they had a different energy, and performed a lot of the songs very differently, so think vhs growltiger would probably have been a whole different cat than broadway growltiger and it's a shame we never got to see that.
i also eventually came to own this exact same recording on cd, and the vhs film on dvd. for futureproofing, mainly. the more often you watch a vhs the more degraded the tape gets.
it's all gone now, but there used to be a lot of little CATS fansites and fan-forums and communities online, and they were my first experience of fandom, albeit one i only ever observed and did not participate in. also my first exposure to fanfiction that i hadn't written, and thus to the word itself.
or if the CATS broadway recording doesn't count, by virtue of technically not being an album, then the first album i ever bought for myself was
2)

enchant by emilie autumni could talk about ea for hours uninterrupted, but i won't. i own all but the latest of her albums, some of them twice over, along with two different versions of the asylum. some of her albums are on both of my mp3 players, despite the fact that i mainly carry two for the sake of space. ea's lyrics, music and attitude have massively influenced my approach to both art and self-expression in general. in particular, her declaration that she often dresses so brightly to show goths that they can in fact wear colour was how i justified to other goths my own embrace of colour. i used her music as a proxy to experiment with and then embrace femininity. i drew a heart on my cheek for years and have considered more than once having it tattooed on. her music both got me into and later out of the most horrific experience of my life, not in a metaphysical sense but in a literal one as a result of the connections that music caused me to form with fellow listeners.
laced/unlaced (well, the laced side) is my favourite album of hers, and i'm still sad that we never got that much violin from her again.
her accompanied poetry is worth listening to if you don't like her musical styles, and i still think about ghost regularly.
The Year 3000 by Busted
My introduction to Pop punk, even if it was an industry plant band 
i've always wondered what it meant for a band to be an industry plant, but i just read busted's wikipedia article, and:
The band was launched in August 2002, making their first appearance on the cover of Smash Hits with the headline: "Meet Busted: They're Going to Be Bigger than Rik Waller!", making it a first for any pop band to appear on the magazine's cover before releasing a single.
i no longer have any questions about what "industry plant" means.