In the Doctor's timeline, their first meeting was in the Library ("Silence in the Library"). In River's timeline, their first known on screen meeting was when the young Melody Pond (River) was in the spacesuit in Florida (at the end of "The Impossible Astronaut").
Note that, although they seemed to meet in "A Good Man Goes to War", shortly after Melody's (River's) birth, that was a ganger (Flesh duplicate) of baby Melody. In any case, even if the real baby Melody was aware of that meeting, through the link with her ganger, she'd be far too young to remember it. At the end of the episode, the Doctor says he is going to search for Melody, however it has yet to be revealed whether he ever actually located her as an infant, or if the above encounter was actually their first.
The videogame The Eternity Clock makes things even more complicated by revealing that River has actually met all of the Doctor's previous incarnations, but erased their memories of the encounters. If the BBC-licensed games are considered canon, then the Doctor actually first met River during his first incarnation but only retained the memory from the Library encounter onwards.