Homebrew is accurate - it can't be anything but a cable system problem. An entire replacement of the cable and housing is indeed a good idea, but if you have been unable to diagnose the problem yourself you may want to have a shop do the work to make sure the housing is cut, sized, terminated and seated properly, that the cable is lubricated if necessary and that the derailleur is properly adjusted.

If you want to do the work yourself I would suggest you start by familiarizing yourself with how the lever, cable system and derailleur work together. When you shift to larger cogs the lever pulls on the cable, which then moves the derailleur inward. When shifting outward to larger cogs the lever releases cable, which then has to be pulled through the housing by the derailleur using spring tension. If there is too much friction in the cable (or rarely if the spring is weak/broken) the derailleur cannot easily pull the cable through. Of course the system works the same for the front, but in most cases the derailleur moves outward when the lever pulls, inward when it releases.

It's fairly easy to isolate the problem. Shift to the largest rear cog and then, without pedaling, shift the lever to the smallest cog position. The cable will now be slack. Using your hands pull the cable back and forth on either side of a section of housing. If you have brifters (shift/brake levers) you will have to shift the lever back and forth while you hold onto the first section of cable. You should be able to detect where the friction is high. You may also see rust on the cable when it emerges from the housing. Friction may be caused by rust or dirt where cable passes through housing, kinked cable, distorted housing ends or cable stops, or broken cable strands. If the cable is rusted at all always replace both cable and housing. Study several tutorial/video on cable replacement (just Google replace rear derailleur cable) before you decide wheter to do the replacement yourself, and you will need good cable/housing cutters to do a neat, effective job.