A geared push bike (or bicycle) will feature some sort of device for shifting gears. The hand (or "twist") grip gear shifter is one such design. The shifter employs a rotating sleeve, which is "gripped" by the inside of the hand and twisted in one direction or the other (depending on choice of gear). Hand grip gear shifters will operate poorly if grit and grime have accumulated inside the mechanism or if grease on internal contact points has dried out.
Rotate the shifter sleeve so that the chain shifts to the smallest front chain ring or rear cog. If fixing the shifter on the right side of the handlebar (which controls gears at the rear of the bike) shift to the smallest rear cog. If fixing the shifter on the left side of the handlebar (which controls gears at the front of the bike) shift to the smallest front chain ring.
Follow the cable from the shifter to its terminus. Use an Allen key (also called a hex key, due to its hex-shaped head) to loosen the bolt anchoring the cable at this end. If there's a metal crimp at the tip of the able, clip it off with wire cutters.
Pry open the cable hatch on the body of the hand grip shifter. Using needle-nose pliers, pull the cable free of the shifter. Set the cable aside for now.
Slide the rubber grip beside the shifter off the handlebar. Using the Allen key, loosen the clamp bolt that secures the shifter to the handlebar.
Pinch the metal retaining clips inside the shifter with the needle-nose pliers. While pinching these clips, separate the rotating sleeve from the shifter body. An internal spring will come away with the shifter body.
Use a plastic-friendly cleaner/degreaser and rag to clean the internal parts of the shifter. Wipe away any dirt and grime.
Apply a thin layer of fresh grease to all contact points inside the shifter. This is any point inside the shifter where two parts come in contact with one another.
Slide the rotating sleeve partway onto the shifter body. With the sleeve in this position, reconnect the shifter body spring to the rotating sleeve (there is a pin inside the sleeve to which the end of the spring attaches).
Bring the two ends together and lock them in place. Snap the retaining clip back into the end of the shifter.
Slide the hand grip shifter back onto the handlebar in its former position. Use the Allen key to tighten the clamp bolt, and reinstall the rubber hand grip onto the end of the handlebar.
Apply a small amount of fresh grease over the length of the cable.
Reinstall the cable, inserting it into the shifter and pulling it all the way through, until the head of the cable is seated inside the shifter. Close the cable hatch.
Remove all slack from the cable by pulling the end taught with needle-nose pliers. Using the Allen key, tighten the bolt that anchors the cable at this end (from Step 2 of "Disassembly") onto the cable.
Fit a new cable crimp onto the tip of the cable, and use the needle-nose pliers to mash it in place.