Components of a song
 

As far as musical instruments go, I've only ever played the piano. I never bothered learning about musical theory so I'm completely lost in that area. I'm also not very good at it. I'm trying to write some songs though and could use some advice.
I find that it's pretty easy to listen to a song and to find the melody on the piano. I find that it's harder to pick out the right chords that are playing in the background, but it's still possible given enough time, and the fact that they tend to repeat.

However, when I'm writing my own songs, I can only ever really pick out the melody. And in the event that I do find a chord progression, it just sounds like the melody but with more notes.

What is the objective of the chord progression? How do artists find the right chords for a song given the melody?

To further complicate things, a lot of songs seem to have multiple melodies played by different instruments. It seems that the dominant melody always goes to the vocals but how do they know that these will sound good together? How do they come up with a second melody to the first one.

Thanks.