Back in my sub-contractor days, here's how I put it:

"90 percent of plumbing is real easy.  The other ten percent is why I get the big money."

In general, your standard medium-grade faucet falls in the first category.  It's almost embarrassing when you see someone real good rebuild one in five minutes while talking incessantly the whole time about last night's baseball game and his teenage kid's crazy new hair-do.

Whoosh!

I used to amaze people all the time.  Some housewife would still be out in the kitchen making me a cup of coffee, I'd walk in and she'd say, "So, how's it look?", only to find out I'd already finished.
 

LOW WATER PRESSURE

If both the hot and cold sides are low, it's the aerator on the end of the spout.  I know it seems counterintuitive, because it seems like it'd be like placing your thumb over the end of a garden hose and water would be spraying everywhere, but such is not the case.  Grab your vicegrips, put some paper towels or some cloth around the jaws so you don't scratch the aerator, and spin that baby off.  Take it apart and clean out the grains of sand in the tiny holes with a pin or needle.

If the water pressure is only low on one side, make sure the shut-off valve beneath the sink is fully open.  Compare the number of turns to the other side.  Assuming it's open, and the problem doesn't exist anywhere else in the house, it's probably the faucet valve on that particular side.  Take it apart, look for anything obvious.  The plastic valves usually have a small tang or something to keep them in the correct position, so it might have broken and slipped.
 

DRIPPING

Three out of four times it's the hot side.  The hot water tends to eat up the rubber parts faster than cold water.  If the cold water side seems to work just fine, don't touch it.

Here's the standard routine.  Your mileage might vary:

Turn off the water under the sink.  If there's no valve, there's usually a main valve somewhere around the outside of the house, and there's always a master cut-off valve out by the street, although you'll need a special 'water wrench' to turn it.

  

Pry the little plastic cover off the top of the knob.  Unscrew the little bolt.

  

If the faucet's really ancient and crusty and has metal handles, you'll probably need to pry on it from below with a couple of screwdrivers.  There's also a special faucet handle-pulling wrench at the store.  Rock it side to side in all directions, then give it a pull.

  

Grab the crescent wrench and unscrew the valve.

  

Unscrew the bolt holding the faucet washer in place.

  

Replace the faucet washer and reverse the above process.

 
The problem with dripping faucets and showers is usually the faucet washer, but not always.  If you replace the faucet washer and it's dripping again in 6 months, it's because the little brass piece that the faucet washer is pressing against has a rough edge and is grinding it down.  If it's corroded enough that there's an actual cut in the valve seat, just replacing the washer won't stop the drip in the first place.

There's a fairly cheap tool at the hardware store for removing valve seats, and there are only a few sizes of them, so pick up a new one and you're good to go.

In older faucets (and even some newer ones), the valve seats aren't removable.  The store sells a 'valve seat grinding tool' that you can either operate by hand or use with a drill gun and this'll grind the valve seat down until it's smooth again.
 

REPLACING FAUCETS

How easy or hard this is depends on (1) how old the faucet is and (2) whether or not it's been leaking around the base.  If it's an older faucet with metal nuts, and it's been leaking for a couple hundred years, the nuts may be so corroded you can't even get a tool on them.  You'll have to use a long drill bit and drill them out from below, in a real awkward position with metal shavings flying everywhere.  Not fun.

A newer faucet with plastic nuts will probably pop right out, not even needing a tool for the nuts.

As far as disconnecting the supply lines, you might need a special "basin wrench" to reach up behind the basin. 

Make sure you place a nice healthy bead of caulk around the base plate — it's easy to clean up later and you definitely don't want it leaking up there.