Why doesn't Amy Pond try to find Melody / River Song?

Because Steven Moffat wanted the Doctor to take charge in the search for Melody, presumably as part of the default 'I like flirty, spunky, but when it counts, passive, women' position he tends to fall back to when he's not thinking about it. read more

She did try to find Melody out until River told them who she was. She knows that she Melody was taken away from her and grows up to be River and hence she knows that in the present scenario, Melody turns out to be a really great person (or a Time Lord) when she grows up. read more

Professor River Song (born Melody Pond) was a "child of the TARDIS" and the wife of the Doctor, specifically his eleventh incarnation. She was mostly human, with some Time Lord DNA, conceived by her parents, Amy Pond and Rory Williams, aboard the TARDIS as it travelled through the Time Vortex, and then raised and conditioned by the Silence. read more