Every time I hear that hiss of escaping air from a tire I hope it is not mine. Unfortunately sometimes it is mine and I need to fix yet another flat. Changing a flat is a nuisance at best and a dread at worst. Over the 30 years I've been riding I've developed a system that I will impart to you now.
If the flat is the front tire then release the brake utilizing appropriate quick-release system that accompanies your bike. Release the tension that holds your wheel to the frame by opening the quick-release lever. If you have tabs on your forks, affectionately called Lawyer Lips by some, you will have to unscrew the quick release a few turns in order to remove your wheel. If you do not have those tabs then the wheel should drop out without having to unscrew the quick release.
If the flat is the rear tire then the first thing to do is to lift the rear wheel off the ground turn the crank and shift until your chain is on the smallest cog in the rear. This helps tremendously when remounting the wheel. If alone and once the wheel is off the bike lay the bike down on the non-drive side. Sometimes if a tree with low branches is around suspend your bike by hooking the nose of the saddle on the branch.
Once you have the wheel in your hand look over the outside of the tire for anything obvious that caused the flat. If identified remove the dastardly thing; if not then proceed as follows. Let any remaining air out of the tube then using tire levers start opposite the valve stem and insert the "scoop" portion of the tool under the tire bead. When tire bead is exposed hook the tool onto a spoke.
Using another tire lever, move one or two spokes (about 3-4 inches) in either direction then repeat the process. By the third lever you should be able to move one of the tools around the rest of the rim and release that side on the tire. There is no need to remove the entire tire. Remove the tube and even if you have already removed the probable cause earlier check the inside of the tire using your fingers to identify any protruding object.
If you are not alone ask another person put a few pumps of air into the new tube to give it some shape. Once the item that caused the flat is removed place the new tube into the tire. Place the valve stem in first and release the remaining air, then starting opposite the valve stem start reattaching the tire to the rim. Once past the valve stem exercise the stem into the tire to be sure it is fully inserted into the tire. When the going gets tough ask for help (just kidding!). Using your thumb to hold the tire in place and the fleshy part of your palm just below your fingers roll the tire onto the rim. Try not to use tire tools to remount your tire, in doing so you may puncture the new tube.
Once you have the tire mounted on the rim put in a couple of pumps of air then check to see if the tube is caught underneath the tire bead. You should be able to see the rim tape around the entire circumference of the wheel. When inflating the tube using a normal bike pump place your index finger over the front of the pump and your thumb over the tire. This keeps the pump stable and helps to prevent jostling the valve stem too much. Place your palm at the back of the pump and start pumping. Inflate to full pressure and remount the wheel onto the bike. Close the brake release, exercise the brakes, and make sure the wheel turns freely. If it's the rear wheel shift back into an appropriate gear before starting off.
An after thought; When ever you put your tire on the rim place the tire label inline with the valve stem hole. This way you always know the orientation of the tire and rim interface. If you cannot find the cause then you can put some air into the old tube and find the puncture. Then you can locate the area on the tire where the cause of the flat occurred.
Thank everyone for their help and now it's your turn to take a pull at the front!