Changing a faucet is a simple task that you actually should do once a year, to avoid having it ever drip. The general concept is easy enough, but here are some useful working tips:
Put a plug or strainer in the drain first, to catch screws and other little parts. This avoids having to extend the plumbing job to one of taking the drain apart to get that dropped screw.
Turn off the water, of course. If you don't have shut-off valves under the sink, it might be worth installing them to make the job easier next time.
The little plastic lid on the handles, which usually indicates Hot and Cold, can be pried off with a knife.
Remove the retaining screw (always a medium size cross type screw driver).
Now you need to remove the handle itself. If it won't just pull off with a little wiggling, check out Working the Handle Off, or Faucet Handle Pullers.
Now you need to unscrew the retaining screw that is holding on the washer. The problem is how to hold onto the faucet stem while using the screwdriver. Simply place the stem into the faucet handle. That will allow you to hold the much larger handle with one hand while unscrewing the retaining screw with the other.
Match the new washer to the old one. You don't want it too tight into the seat, nor rattling loose. For some fascinating information on washers, check out Washer Numbers.
One more very important detail before you put it all back together. Hold the sleeve of the faucet stem and twist the handle as to open the faucet, like you want to turn on the water. This will draw the shaft of the stem back into the sleeve. If you don't do this, you will have a real problem screwing the stem back into place, because the rubber washer may be bottoming out in the faucet, preventing you from making a leak proof seal with all the rings that hold the stem in place.
The hot water washer always gets hard and leaks before the cold, but while you are at it you might as well replace both. If it still leaks after all of this, you may need to replace the 'seat', which is down in the bottom of the faucet. It is called the 'seat' because this is where the rubber washer 'seats' to close off the water. If you can get it out, take it to the store to find a match. You will need a 'seat wrench' to get it out, and on some very old faucets, it simply won't come out. Then it is in fact time for a new faucet.