Rear gears will not shift

[Banjopickr]

I got a 15 speed Murray bike via garage sale. I cleaned it up got the breaks working well, but it looked like it had been left out in the rain. My problem is with the gear shift. My derailing shifter works fine ( it will shift from 5-10-15 no problem, but the other shifter that shifts the gears on the the back wheel, 1-2-3-4-5, does not shift. When I move the shifter the cable stays in the same position on the handle bar. Its like the mechanism that is supposed to retract the cable down towards the bottom of wheel is not doing that. Unfortunately I do not know how the mechanism at the back wheel works, or what it is called, so I do not know how to resolve the problem. I did trace the cable to the back wheel and made sure that it had movement, but when pull the shifter towards me, I can see the end of the cable sticking out by the shifter on the handle bar, like the cable isn't being pulled from the back wheel assembly. Can you give me some insight to how that assembly works and how to fix it? Thanks

[Skyguy9999]

As you said, it has been left out in the rain. I wouldn't be surprised if there's some rust within the cable & housing that leads to the rear derailleur.
I'd start by disconnecting the cable from the rear derailleur and then tracing it through to check for movement. More or less, run through the gears on the shifter and see if the cable moves. If it doesn't then you would need to look into freeing the cable by shifting the cable housings around (spray a little WD-40 or Liquid Wrench into the end of the housing, first).
If this works and frees the cable, you can then proceed to <a HREF="http://bikeride.com/lube-brake-shift-cables/">lubricating the shift cables</a> around the cable housings and making sure you can slide each cable housing, easily. Of course, if you see the cable that was within the housing and it looks rusted, you might look into <a HREF="http://bikeride.com/replace-cable-housings/">replacing the cable housing</a> and shift cable. That might be advisable anyway, depending on how much money you are wanting to put into this.
If the cable is fully freed (or you free or replace the cable), then you can move on to <a HREF="http://bikeride.com/adjust-rear-derailleur/">adjusting the rear derailleur</a>.