When purchasing a new sink drain, make sure you buy one that is the proper diameter to screw into the top of the sink trap part of the pipes below your sink and into the base of the sink.
Shut off the water supply by screwing the hot and cold water knobs below the sink clockwise. Put caulk around the drain lid outer edge and screw the drain lid into the base of the sink bowl itself. Get under the sink and put caulk in the threads under the sink. Place the gasket on the sink drainpipe and screw it into the bottom of the sink. Screw the sink trap onto the base of the sink drainpipe.
Grab the faucet drain plug handle that is sticking down under the sink from the backside of the faucet. Connect the drain plug handle to the top of the slotted drain plug adjuster. The slotted drain plug adjuster is connected to the drain plug lever on the sink drai pipe itself. Test the drain plug by pulling the drain plug handle behind the faucet up and down. You can adjust what slot the drain plug lever from the sink drainpipe is in and this will either help or hinder the drain plug to close. Adjust it until it closes completely.
Allow the caulk to dry and set for several hours. Turn on the water supply by screwing the hot and cold water knobs counterclockwise. Test that the sink drain plug works by pulling the drain plug handle behind the faucet up and down. Make further adjustments to the drain plug lever if needed.