Three speed hubs are shifted much like indexed derailleurs, using a
special cable with a barrel attached to either end. It's fairly common
for these cables to seize up with rust, so I'll show you how a new one
goes on. The bike I'll be working on has a Shimano 333 hub with a twist
grip shifter. First you remove the shifter from the bike if it's still attached (on this shifter you unscrew the two screws on either side):
Now you can disassemble the shifter, just don't lose any parts (especially the ball bearing that controls the shifter's indexing):
Now install one end of the cable onto the shifter's core piece:
Let the cable sit inside the channel:
Reassemble the shifter around the cable, paying attention to where the ball bearing sits:
Screw this barrel into the shifter once the two halves are back together:
Now you can slide the shifter onto the handlebars. You should tighten the clamp screws so that the shifter stays in place. This shifter also has a third smaller screw on the bottom to adjust the twisting resistance. You can adjust this screw after the shifter is secured on the handlebars:
Now you have to secure the other half of the cable to the frame. This bike uses a removable clamp to hold the cable in place. Basically you need to find a position that takes most of the slack out of the cable once you screw it onto the hub's pushrod lever mechanism:
Once you find a good position for the clamp you can tighten it down. Here's an overview on how to adjust the shifting:
There will usually be a red circle or a pointer arrow on the lever that the cable attaches to. This arrow needs to line up with the indexing notch when the shifter is in the middle gear. You can use the barrel adjuster to fine tune this adjustment. If you don't have enough threads on the adjuster to line up the arrows, then reposition the chainstay clamp to give yourself more room to adjust the cable. That's about it. These hubs usually have an oil port, so fill it with a little oil every now and then (gear oil would be just about perfect, but motor oil or automatic transmission fluid will work well too).