Remove kitchen sink plumbing
Without pictures it will be hard to offer anything but very general advice.

Here are things to consider:


There should be shut off valves under the sink for the hot and cold water. You will want to keep those in place and turned OFF before you start removing anything. If it is necessary to replace these valves you will have to shut off the water to the whole property for the duration of when you take out the old valves and install the new ones.
The water lines will have fittings or compression nuts that can be unscrewed to remove the lines from the shut off valves to the sink faucet. 
The sink faucet may come out directly with the sink or need to be removed separately. It will depend if the faucet mounted to the sink itself or to the counter top.
The drain line pipes and the p-trap are removed by unscrewing a series of large diameter fittings that will find at the various joints in the piping. 
Once the drain is removed all the way up to the pipe that sticks out of the wall you will want to make sure to temporarily seal off that pipe to prevent sewer gas from coming into the room. One way this can be done is to stretch several layers of plastic bag over the end of the pipe and secure in place with a heavy rubber band looped a couple of times over the end of the pipe.
Sinks generally mount either by hanging through a hole from the top of the counter top or up from underneath. In most cases there will some type of clamping brackets under the counter top around the periphery of the sink that hold the sink in place. Remove these to allow the sink to come out. The type of clamps used (if any) varies greatly. 
Some sinks will have been sealed to the counter top with some type of caulk or silicon sealer. This may have to be broken loose to remove the sink. In some cases it may be easier to just cut the counter top around the old sink to remove the thing - depending upon the material used for the counter top.