First LP I personally went out and purchased with my own (hard-earned lawn mowing) funds: The Sound of Silence by Simon and Garfunkel. When it was new... what, 1966 or so?
To be completely honest, that's a slightly false answer, because I clearly remember buying several at the time. But for the life of me can't remember what they were. So SOS walks away with the prize by virtue of being memorable.
Interesting thread... who would have thought, out of all the important firsts in most people's lives, that buying the first record would be so easy to recall?
-Glenn