Disconnecting the Plumbing
Turn off the water valve underneath the sink. If it is not there, you might have to turn off the main water supply to the house.
Disconnect the p-trap underneath the sink using the adjustable pliers.
Place a plastic pan underneath the sink and remove remove the p-trap by unscrewing it from the drain pipe.
Disconnect the water supply lines that feed the faucet from the wall. Then you can remove the sink and faucet from the countertop completely.
Removing the Countertop
Place plastic sheets on the floor, in doorways and in any areas where you do not want debris and dust to fall. Tape the plastic down so that it does not come up while you are working.
Pry the backsplash from the wall with a crowbar and hammer. Do not rush this as you can damage the drywall.
Once the backsplash is removed, you can start to remove the countertop. Tile countertops will need to be chiseled and pried out a little at a time. You also can try to grind down the grout lines to lift up the tiles. If you have a power chisel you will be able to make this work go by very quickly.
Installing the New Countertop
Measure the area for the new countertop. Decide if you are going to reuse the old sink or by a new sink before you purchase your countertop as they will manufacture the countertop based on the sink you are using.
You may need to reinforce the cabinets below the new countertop you are installing. Some older cabinets may not be able to take the weight of a granite or marble countertop, which could weigh significantly more then an old tile countertop.
Run a bead of silicone along the top of the cabinets and any area on which the new countertop will rest.
Carefully place the new countertop on the cabinets, making sure not to smear the silicone. Press down firmly once the countertop is in place. You may need help lifting the countertop pieces, especially if you are installing granite or marble.
Install the sink and reconnect the plumbing per the manufacturers directions.