In 1963, my cousin Judy gave me her little RCA Victor 45 player. The first record I bought was "I Will Follow Him," by Little Peggy March. That Christmas, my parents bought me my first stereo, a suitcase model from Motorola, plus two Beach Boys albums (Surfin' USA and Surfer Girl), two by Lesley Gore, and one by Skeeter Davis.
One reason I got a stereo for Christmas was because my mother got so sick of my playing "Be My Baby" on repeat during assassination weekend.
My first album I chose for myself was Meet the Beatles in January, 1964.
I still like playing songs on repeat. iTunes and YouTube are good for that.