At the end of 15 minutes, cut the heat. Now is the time to cool. You want to cool the wort as quick as possible to <100 degrees F, preferably <80 degrees F. The faster you are able to cool, the more pronounced the cold break will be. Like the hot break, this is another point where the temperature causes protein to coagulate, and more coagulation is better. The difference is that you won’t have to worry about boil-overs, though, which is quite nice. I learned that cooling 3 gallons of liquid from 212 degrees F to 80 is actually rather difficult. The most basic method is creating an ice bath. I took a hand towel, folded it, and covered the sink drain with it. I used a hand towel primarily because I didn’t have an actual stopper, and secondarily because it slowly drains water. I put the lid on the hot brew pot and place it in the sink on top of the towel. Now I turn on the sink for water and start adding ice. To do it without worries, the 3 gallon cool will require about 40 pounds of ice. Yea. Just start adding the ice to the sink around the pot and let water keep filling it. Repeat until a) out of ice and b) you have a brew pot that is cool to the touch at all points. One modification is to only use water at first and then begin to use ice about halfway through. This is a physics/chemistry thing. Basically, room temperature water will have a similar cooling ability as ice water on a 212 degree pot, but as the temperature of the pot approaches that of the water, water will become much less effective at cooling while ice will remain useful. So you can get away with using less ice this way, which is nice.
There are other, better ways to cool the pot. Another way is to buy an immersion chiller. This is a coiled copper tube that you put in the wort at the end of the boil (last ten minutes or so, to sterilize it). A hose attached to one end of the copper tube attaches to the sink faucet, and a hose attached to the other end can be placed in the drain. This is by far the easiest and most effective way to chill your wort, especially when combined with a pre-chiller, which is essentially the same thing outside of the wort that will further cool the water before it goes in. The problem with these methods is that I don’t really feel the need to drop $60-100 or more on a piece of copper tubing. It is admittedly cost effective if you would otherwise be buying ice from, say, a gas station and plan on brewing ten or more batches of beer. I just don’t have that cash to drop right now. These chillers are a ‘set it and forget it’ solution, definitely a luxury. It will decrease your wort chill time down from 30-45 minutes to 10-15 minutes and you don’t need to baby it, so you’re free to do other things (like clean up).
The cooling stage is one where you are more likely to infect the beer with unwanted bacteria. The cooler the wort gets, the more desirable a home it becomes for bacteria. Keep in mind it is warm and already full of water and sugars, which is exactly what they want. I have read about methods where you freeze water in tightly closed plastic bottles, sanitize the bottles, and then add them to the wort, using the bottles as giant ice cubes. I recommend against this method because the efficacy would not really outweigh the risk of contaminating the wort with foreign bacteria and such. You can also do your cooling with the lid off. The steam action will definitely facilitate quicker cooling. Again, you increase your risk of infection this way, but in my experience so far this is a relatively small increase. I often find myself almost knocking the running water into the wort, and having the lid on has saved me every time (tap water does contain trace bacteria). Off the top of my head, these are all the methods I’ve read about. I find this step to be the most difficult/stressful on brew day.
In the meantime, prepare your carboy. Since you started out with boiling 3 gallons of water, you’ll want to have about 2 gallons of water in the (sanitized) carboy. If you have your water well ahead of time, you can cool it in the fridge, and then it will further cool the wort that tricky few degrees between 75 F and 65 F. Once you’ve got the wort cooled to room temperature, add it to the carboy. You’ll be wanting a funnel to do this, and many people also use a filter to keep solids from going into the carboy, but this isn’t really necessary for beginners. Finally, check your volume to make sure you’ve got about 5 gallons, and if not, add more clean water.
One of the important ingredients for strong fermentation is oxygen. Even though fermentation itself is anaerobic, oxygen is used by the yeast to reproduce. I find it easiest to aerate the water in the carboy before you add the wort to it by way of shaking the hell out of it. Just shake the carboy, or pour the water back and forth between (sanitized) buckets. The more the water is disrupted, the more air is getting in. You can shake it up/pour it back and forth after you add the wort, but if you aerate the wort when it is too warm, you can begin to oxidize the beer (which is the last thing you want). There are also other mechanical methods of aerating the wort, but seriously, what is so hard about shaking up three gallons of water? You don’t need an aquarium pump and oxygen stone or any other complicated system. Just do it. If you don’t properly oxygenate your wort, the yeast may run out of what it needs to keep going and so the whole operation just kind of shuts down (a “stuck” fermentation). You don’t want that, so just shake your dang water.
Finally, you’ve got 5 or so gallons of wort in your carboy at room temperature. You’ve also got an activated yeast smack-pack ready to go. Sanitize the pack and a pair of scissors, cut it open, and pour the yeast into the wort (again, I recommend using the funnel). Put the airlock in the carboy and move it to the cool, dark location where it will sit for the next 2-6 weeks.