River grew up knowing of the Doctor. As was explained in "A Good Man Goes to War", the Silence kidnapped Amy, shortly after she became pregnant, because her child had been conceived in the TARDIS, whilst in flight through the time vortex, which gave the child some (not all) of the characteristics of a Time Lord. The Silence sought to enhance those characteristics, especially the ability to regenerate, while the child was still in Amy's womb. Their intention was to make the child into a weapon against the Doctor. When the child, a daughter, was born, Amy named her Melody. The name she later used, River Song, was a translation of Melody Pond into the language of the Gamma Forest, with given name & surname reversed. In "The Impossible Astronaut"/"Day of the Moon", young Melody managed to escape from the Silence. She regenerated for the first time in New York & then somehow (we don't know how) found her parents in Leadworth. However, when she found them, her parents were still children of about the same age as she appeared to be. She grew up with them, hearing Amy's stories of the "Raggedy Doctor". Eventually, as a young adult, she met the Eleventh Doctor & travelled with him, Amy & Rory to 1938 Berlin ("Let's Kill Hitler"), where she regenerated again -- into the incarnation usually known as River Song.
Because River's timeline is shown in a very scrambled order, it can be difficult to follow but the four episodes mentioned above (all in Series 6) cover most of what we know of her early life. Some additional details are given in "The Wedding of River Song", the Series 6 finalé.