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Carefully scrape around the faucet and the wall with a putty knife or a razor blade and lift up as much caulk as you can. Continue to scrape away at the caulk, removing small portions, until there is no more caulk on the wall and faucet. If your faucet doesn't have caulk connecting it to the bathroom wall, you can skip this step. Grab the faucet with both hands and turn it counterclockwise. Unscrewing the faucet counterclockwise will loosen and remove it from the wall. As their name implies, screw-on faucets are screwed onto metal threads on your water supply pipe. Open the handles on an adjustable wrench and carefully place it around your faucet. Tighten the wrench around the faucet and slowly rotate the wrench in a counterclockwise motion. Rotate the faucet 2-3 full rotations. This should loosen it up enough so that you can unscrew it with your hand. If you're having issues turning the faucet, you can use a screwdriver as a makeshift lever. Stick the screwdriver into the hole in your faucet and push it up to turn the spout counterclockwise. Once it's loose, unscrew it with your hands. Once you've turned the faucet counterclockwise 4-5 times, you can simply slide it off of your water supply pipe. If you don't need to remove an adapter from the pipe, you're done. An adapter will look like a threaded metal piece that's fit over your water supply pipe. There's a recessed screw on the top or bottom of the piece. Remove the screw by turning it counterclockwise with a hex key or Allen wrench. Once the set screw is removed, the adapter should slide on and off the water supply pipe easily. Take the adapter off the pipe to complete the faucet removal.
Remove the caulk from around the faucet, if applicable. Turn the spout counterclockwise. Turn the faucet with a wrench if it's stuck. Use a screwdriver if you don't have a wrench. Pull the spout from the plumbing. Remove the set screw in the adapter if you have one. Slide the adapter off of the water supply pipe.