Doctor Who: Every Character Who Can Fly The TARDIS
7. River Song
From her very first appearance in Silence In The Library, it was made abundantly clear that River Song was no ordinary acquaintance of the Doctor's. It was heavily implied that the two had a deep, personal connection, and this is something that was slowly explored over the next several years of the show.
Along the way, we learned her true identity, we found out how she got her sonic screwdriver, and we were also shown that she can pilot the TARDIS quite capably, with no input required from the Doctor himself.
In fact, River seems to know more about the TARDIS than the Doctor does. In Series 5's The Time Of Angels, she shows him how to shut the brakes off before landing, which, to his immense disappointment, means that the ship no longer makes its famous "wheezing" noise.
As for how River learned to fly the TARDIS in the first place... well, that's a little complicated. In the Series 5 finale, she tells the Doctor that he taught her, but in Series 6's Let's Kill Hitler, she claims that the TARDIS itself showed her the ropes (which does make a lot of sense, considering that River was conceived in a TARDIS bedroom).
So, maybe it's a bit of both - the TARDIS taught her the basics when she desperately needed them, but the Doctor gave her some advanced training further down the line.