Summarize this article:

Remove any personal items and other shower accessories from the bathtub and place them somewhere out of the way. Wash the tiled area with the bathroom tile cleaner. Find a cleaner that will get rid of soap scum without leaving a residue. You can also use a mild dish detergent and hot water to clean the tiles. Use the utility knife or razor to make an incision on one side of the caulk seam. Hold the knife so it is near the wall at the base of the silicone and slide the knife down the full length of the seam. Slice slowly and be careful not to cut into the wall. Don't cut all the way through the seam. Your goal is just to loosen the edge of the seam. Make a shallow cut by using only the tip of the knife. Repeat the previous step on the other side of the same seam. Slide the knife along the length of the seam close to where the silicone touches the tile, but again without slicing into the wall. Peel the caulk up and away from the tile. This will remove the silicone that was filling the joint, along with the part you can see. If you encounter any resistance from the sealant, use the putty knife to push it along. Use the utility knife or putty knife to dig out any leftover pieces of silicone carefully. Position the knife at an angle to the tile and take your time to avoid scratching or damaging the tile. Repeat the steps for any other seams that you want to remove. Take your time and continue to work carefully. Wet the scouring pad with acetone and wipe it over the bathroom tiles. It may take a little elbow grease to get rid of tougher residue. If you don't have acetone, you can use rubbing alcohol or mineral spirits instead. Use a mixture of a cup of bleach and 1 gallon of water to kill any mold or mildew. Wait until it dries completely before adding new sealant.

Summary:
Clean the shower or bathtub. Select the first caulk seam to remove. Hold one end of the loose silicone sealant. Remove the remaining sealant in the joint. Scrub the tile to remove any residue.