The very first thing we'll do is loosen the lug nuts on the affected wheel. The reason we do this is because loosening the lug nuts requires some pressure, and it's safer to do this while the vehicle is not jacked up in the air. We'll loosen each lug nut only slightly.
Do not remove the lug nuts from the wheel at this point.
Next, we'll need to remove the wheel, and to do so we must jack up the vehicle near the affected wheel.
Most vehicles have designated "jack points" near each of the four wheels. Use your owner's manual (or Google) to locate the jack points. Place the jack underneath the jack point, and raise the vehicle so that the tire is a few inches off the ground.
This guide is not primarily about how to remove and replace a wheel, but I will provide some basic instructions. Ideally, you've done this before and have some confidence in the process. If not, please seek additional guidance.
Since we've previously loosened the lug nuts, we should be able to easily remove them. You can use the T bar, or remove them by hand, if possible.
Once the lug nuts are removed, slide the wheel straight out off of the lug nuts, and move it to your work area.
There are two options here. If you already know where the puncture is (e.g. there's a clearly visible nail or screw sticking out of the tire), you can skip to the next step. In many cases though, you won't be able to find the hole easily. Luckily, there is a solution: soapy water.
Prepare a large pitcher with soapy water. With the wheel removed from the vehicle, cover the wheel entirely with the soapy water. Slowly examine each part of the tread, rolling the tire as necessary. You're looking for new bubbles rising from the surface. A fast leak will be easier to identify. With slow leaks, it might take some time.
See the related picture for an example of what this can look like.
Before moving on, we'll need to remove the debris, if necessary. To do this, you can use whatever tool you deem appropriate, but in most cases a pair of pliers should work.
With the puncture location identified, we now need to prepare the hole. Use the rasp tool to clean out the hole. This will do two things. First, it will remove any additional debris. Second, it will open the hole slightly to more easily insert the plug.
Insert the rasp tool into the hole, and quickly move it up and down. Do this for ten or fifteen seconds, or until you feel consistent resistance with each cycle.
At this point, I typically leave the rasp tool in the tire until I'm ready to insert the plug. This prevents the tire from losing more air than necessary.
Find the insert tool and a string plug. The insert tool is forked at the end, so slide the string plug through the opening like you're threading a needle. With the string plug pulled through about half way, add some rubber cement to the plug.
Here's the fun part. We need to push the insert tool into the hole. This often requires some effort, and you need to ensure the plug isn't pushed in too far. The goal is to insert the plug two thirds (2/3) of the way in. Use the picture as a reference.
Now we need to remove the insert tool. To remove the tool, grab the T handle and pull straight out, with one swift motion. If done correctly, the insert tool will come out easily, and the string plug will be left in place.
Again, this step requires experience. Seek guidance if you're not comfortable putting your wheel back on. I would hate for you to watch your wheel roll past you on the freeway.
In any case, at this point you secure the wheel back onto the vehicle. Use a torque wrench to tighten the lug nuts to the proper tightness.
And finally, fill the tire up to the correct pressure. You can find the correct tire pressure in the owner's manual, on the tire itself, and often on the inside of the driver's door.
At this point, you've finished! Enjoy your not-flat tire.