Measure the distance between the hot and cold knobs on the existing faucet. If your faucet only has one knob, then the measurements will be done from under the sink.
Look under the sink, measure the distance between the holes in the underside of the sink. Note where the water connects.
Shut off the water at the shut off valves under the sink. If you don't have any shut off valves under the sink, then turn off the water at the water meter. Be sure to tell everyone in the house that you are doing this.
Remove the hot and cold water hoses with your adjustable wrench, as well as anything else that might be connected to your faucet, such as a sprayer.
Measure the diameter of the water hoses.
Note the thread size of the water hoses, it will either be coarse or fine. Fine threads will take a very long time to disconnect, whereas coarse threads will only require a few turns. This is important to the fit of your replacement faucet.
Measure everything again. Measure twice, shop once.
Shop for a new faucet. Be sure to get one that will fit in the space provided.
Remove the old faucet. It should just pull right out. If not, you may not have removed all of the connections under the sink, check to make sure. Some faucets have an extra lock washer holding the unit against the bottom of the sink.
Place the new faucet into the spot you removed the old one from.
Put a small amount of teflon thread tape on the faucet side of the threads. Once around the prong should be enough.
Reattach the water hoses, and anything else you may have taken off, like the sprayer. Make sure everything is really tight.
Turn on the water and check for leaks. If you have leaks, turn off the water and tighten the hoses again. Do this until the leaks stop.
Try the faucet. Turn it on slowly at first, then if there are no leaks, turn it on to full. Do this with the hot and cold water. If there are no leaks, you are done.