To remove the old faucet, start by clearing out the cabinet under the sink and laying down towels. Turn off the hot and cold stop valves and open the faucet to make sure the water is off. Detach the sprayer hose from the faucet sprayer nipple and unscrew the retaining nut that secures the sprayer base to the sink deck. Pull the sprayer hose out through the sink deck opening.
Spray the mounting nuts that hold the faucet or faucet handles (on the underside of the sink deck) with penetrating oil for easier removal. Let the oil soak in for a few minutes. If the nut is rusted and stubbornly stuck, you may need to drill a hole in its side, then tap the hole with a hammer and screwdriver to loosen it.
Unhook the supply tubes at the stop valves. Don’t reuse old chrome supply tubes. If the stops are missing or unworkable, replace them. Then remove the coupling nuts and the mounting nuts on the tailpieces of the faucet with a basin wrench or channel-type pliers.
Pull the faucet body from the sink. Remove the sprayer base if you wish to replace it. Scrape off old putty or caulk with a putty knife and clean off the sink with a scouring pad and an acidic scouring cleaner. Tip: Scour stainless steel with a back and forth motion to avoid leaving unsightly circular markings.
HOW TO INSTALL A PULLOUT KITCHEN SINK FAUCET
Install the base plate (if your faucet has one) onto the sink flange so it is centered. Have a helper hold it straight from above as you tighten the mounting nuts that secure the base plate from below. Make sure the plastic gasket is centered under the base plate. These nuts can be adequately tightened by hand.
Retract the pullout hose by drawing it out through the faucet body until the fitting at the end of the hose is flush with the bottom of the threaded faucet shank. Insert the shank and the supply tubes down through the top of the deck plate.
Slip the mounting nut and washer over the free ends of the supply tubes and pullout hose, then thread the nut onto the threaded faucet shank. Hand tighten. Tighten the retainer screws with a screwdriver to secure the faucet.
Slide the hose weight onto the pullout hose (the weight helps keep the hose from tangling and it makes it easier to retract).
Connect the end of the pullout hose to the outlet port on the faucet body using a quick connector fitting.
Hook up the water supply tubes to the faucet inlets. Make sure the tubes are long enough to reach the supply risers without stretching or kinking.
Connect the supply tubes to the supply risers at the stop valves. Make sure to get the hot lines and cold lines attached correctly.
Attach the spray head to the end of the pullout hose and turn the fitting to secure the connection. Turn on water supply and test. Tip: Remove the aerator in the tip of the spray head and run hot and cold water to flush out any debris.
Variation: One-Piece Faucet with Sprayer 
Thoroughly clean the area around the sink’s holes. Slip the faucet’s plastic washer onto the underside of the base plate. Press the faucet in place, and have a helper hold it in place while you work from below.
Slip a friction washer onto each tailpiece and then hand tighten a mounting nut. Tighten the mounting nut with channel-type pliers or a basin wrench. Wipe up any silicone squeeze-out on the sink deck with a wet rag before it sets up.
Connect supply tubes to the faucet tailpieces. Make sure the tubes you buy are long enough to reach the stop valves and that the coupling nuts will fit the tubes and tailpieces.
Apply a 1/4" bead of plumber’s putty to the underside of the sprayer base. With the base threaded onto the sprayer hose, insert the tailpiece of the sprayer through the opening in the sink deck.
From beneath, slip the friction washer over the sprayer tailpiece and then screw the mounting nut onto the tailpiece. Tighten with channel-type pliers or a basin wrench. Clean up any excess putty or caulk.
Screw the sprayer hose onto the hose nipple on the bottom of the faucet. Hand tighten and then give the nut one quarter turn with channel-type pliers or a basin wrench. Turn on the water supply at the shutoff, remove the aerator, and flush debris from the faucet.
