Make sure that the shower is dry before you begin working. Use the knife or putty knife to remove the old caulk that you will be replacing. You should be able to scrape the caulk out of the crevices of the shower enclosure rather easily, especially if the caulk is beginning to lift or fall out.
Take care not to scratch the shower pan or the shower while removing the caulk. If you happen to scratch the tub, try using a product like Softscrub to remove the marks. They should come off easily, provided they are just surface scratches. You may also want to try a bit of alcohol on a cotton ball to remove surface scratches.
Use a heat gun to get the caulk off, if you cannot remove it with the knife or the putty knife. The heat gun will loosen the caulk, and make it easier to remove. In order to protect your hands and the shower enclosure, make sure that the gun is set at a temperature below 300 degrees.
Clean the area after you have removed the caulk, using a sponge and water. Do not use soap. If there is any mildew or mold around the area, remove that as well. This can also be accomplished by using just a sponge and water. However, if there's a lot of mold or mildew, you may have to use a mildew remover. Make sure the area is dry before you begin caulking.
Apply just a small amount of caulk to your starting area. It should be no bigger than ¼ inch. Moisten your finger with a bit of water (not too much), and glide your finger along the bead of caulk, smoothing it over the joint, and applying pressure at the same time.
Fill all of the joints the same way. Inspect the joints for any areas you may have missed. Let the caulk dry before you run water in the shower or in the tub. Let it dry at least overnight.