Step 1: Disconnect the battery
Step 2: Find the leak
First step in finding a leak is to get under the car and follow the bright green coolant trail. Unfortunately, in this case the trail quickly hid behind a pulley.
Second step in finding a leak is to just take everything off that might be in your way off. Don't be timid here; if it looks like it might come off, take it off! I ended up taking off the coolant reservoir, wheel, belt, alternator, idler pulley, water pump pulley, etc... And I still didn't find an obvious leak! Though, I did find a suspicious puddle of coolant on the crankshaft pulley.
Step 3: Really find the leak
So, I pressurized the coolant system by running a hose into it. To make a seal between the hose and coolant system, I used an old bicycle inner tube (Anyone who's hung around TimAnderson knows that these are the solution to all of your problems).
The area behind the water pump pulley starting spraying water. My mission was clear: I had to replace the water pump.
Step 4: Drain coolant
Step 5: Remove everything to get to water pump
To remove the serpentine belt, you'll need a breaker bar (i.e. lever arm). There's a notch in the back of the automatic tensioner that you attach the breaker bar too. Pull the tensioner up and remove the belt once it's loose.
Everything else is just removing screws and hose clamps. Be sure to take a bunch of pictures because you'll probably forget where everything goes when you try to put it all back together.
Step 6: Remove broken water pump.
After the water pump is off, scrape off any old gasket that's still hanging on (but again don't scratch the metal).
Step 7: Prepare new water pump
Apply some RTV silicone to one side of the gasket. I'm not sure exactly how much to put on. The internet says that you should not put "too much" or "too little" on, without really going into more detail. Put the gasket on the water pump. Then, apply the silicone to the other side.
Step 8: Install new water pump
Tighten the bolts. Whenever you tighten a bunch of bolts, it's generally a good idea to follow some sort of "star" pattern rather than a "circle" pattern (e.g. top,bottom,left,right rather than top,right,bottom,left).