Goodbye song:
Goodbye (Kelly's Song) by Alabama

Listen to this goodbye song:

Orchestral Country ballad, originally written for the singer's wife as they were separated by travel for a time.
The sound has a poignant and sad feel to it.

Other songs like this one:

You're Gonna Miss This - Trace Adkins
This mid-tempo country song is about the various stages of life, and to make the most of now:
"You're gonna wish these days/Hadn't gone by so fast
These are some good times/So take a good look around
You may not know it now/But you're gonna miss this".

Don't Forget to Remember Me by Carrie Underwood

Another parent/child leaving song, about looking forward to the future, but knowing that the people you're attached to will miss you so much.

You might like:

Into My Arms - Nick Cave

Written and performed by Nick Cave, this is not so much a song about goodbye, but rather about the return of a loved one 'into my arms', wherever they might be.

It is a deeply spiritual song despite possible claims to agnostism in the lyrics.

The above two points are expressed perfectly in the first verse:

"I don't believe in an interventionist God
But I know, darling, that you do
But if I did I would kneel down and ask Him
Not to intervene when it came to you
Not to touch a hair on your head
To leave you as you are
And if He felt He had to direct you
Then direct you into my arms"

Whether you are a parent, lover, or a very close friend, this song will certainly affect you when you hear it, if you are missing someone dear to you.