On most bikes, it's easy enough to free the housing from the slotted frame stops. Shift into low, stop pedaling, shift into high, and you'll have enough slack to pull one housing out. Then pull all the housings off and you can check and service the cable. If time and budget are tight, you can clean (fine steel wool) and oil surface rust. Replace anything that's frayed or deeply corroded. You can inspect the housing ends for failure and crimping. You can check the shifter too, with no load on it.

Cable ends are crimped on with the cable cutter tool. It's often easy to yank them off with pliers if a der or housing needs to be replaced and the cable is otherwise okay.