Now for the fun Stuff:
First of all, you must cut the pipe. Mark where you need to cut with a Sharpie marker or pencil and attach your pipe cutters. Tighten enough to just begin to feel some resistance and start spinning the cutters arround the pipe, Tighten the cutters a quarter of a turn every couple of spins until the pipe breaks apart. Investing in a good Ridgid brand pipe cutter is a wise investment.
Second step is to ream the inside burr left by cutting the pipe. The picture on the right shows two freshly cut pieces of pipe. The pipe on the right has not been reamed, the one on the left has been reamed. You will be left with corkscrew shavings of copper when you ream the pipe correctly. If you fail to complete this step, the water passing through the pipe will experience turbulance once it passes over the burr left on an unreamed pipe. This turbulance can eventually erode a hole directly through the side of the pipe. This translates to a nice size leak.
Above is a video of me soldering two joints. Here are the official steps in heating the pipe and fitting along with some solutions to beginner problems.
1. Heat only the pipe within about an inch of the joint. Heat it in a circular motion to evenly heat the pipe.
2. After about 6 seconds move your flame to the fitting and evenly heat the cup of the fitting.
3. Continue to heat both the pipe and fitting evenly. Flux will begin to bubble out of the joint. Don't worry, keep heating.
**Basically what you're doing at this step is evenly heating the fitting up to roughly 700 degrees (The temperature at which the solder will melt).**
4. Take care not to direct the flame into the joint, as microscopic debris can be thrown into the joint, compromising the whole process.
5. Periodically touch the solder to the joint until it melts upon contact.
6. Once the solder is flowing, push the solder into the joint as you apply it. Continue to heat the fitting cup to give the solder a nice warm place to flow to. You should notice the solder being pulled into the cup of the joint. If it just balls up, one of two things has happenened: you have not heated the fitting enough, or you have heated it too much and burned out all the flux. Either keep heating, or if you're pretty sure you've burned out the flux, dip your brush back into the tub of flux (getting a big old wad on your brush) and quickly apply it to the joint. It will sizzle and smoke but don't worry. Quickly apply the solder as the pipe should be super hot.
7. Once you have applied enough solder (about 3/4" for a 3/4" joint, 1/2" for a 1/2" joint), flick away accumulated solder at the bottom of the joint with a gloved finger and allow the joint to cool before you touch it. If you touch it too early you can crack the joint, in which case just reapply a little heat until the solder re-liquifies, then DON'T touch it.
You should now have a great solder joint. If there is a bunch of burned flux or your joint looks messy, wipe the cooled joint with a towel and use a little sand cloth to polish it up.
With the ability to effectively solder, you can add a water line anywhere you want to; redirect existing pipes, install boilers and radiant heating; etc.