Learn to recognize intervals. When learning a melody, it's good to imagine the melody as a wave or contour that goes up or down. Something like this. Learning to recognize intervals helps you to catch these melodies faster because you know whether the melody note leaps or move in a certain direction as examples. Everytime you learn a melody, analyze the crap out of it, identify the interval & notes used.

It takes time to be able to get used to playing songs by ear so don't worry if you suck during the early stage.

As for harmony, learn & internalize the sound of the chords(major, minor, augmented & diminished). Also learning basic music theory helps when you try to identify chord progression.