Soldering is an art that deserves a whole book, but here's some basic insights.
Use a piece of hardboard to protect the surface you are working on.
Your two (or more) pieces to be joined should be free from visible muck and ready to go.
Almost all modern solder has flux inside it (like writing through a stick of rock), which cleans the wire as it melts onto it.
When you melt solder, a small amount of fumes are given off - this is the flux evaporating. It's probably not a good idea to breathe the flux vapour, so make sure you have adequate ventilation.
Hold the soldering iron as you would a pen, this makes it very easy to manipulate.
Before you try to join your wires, you should 'tin' them first. Read on to find out how...
Apply the tip of the hot soldering iron to the first piece of metal to be joined.
Hold the solder just touching the piece you are heating up - when the piece is hot enough, the solder will melt onto it. Don't melt the solder directly onto the iron if you can avoid it. This piece has now been 'tinned'.
'Tin' all the pieces to be joined.
Hold or lay the pieces to be joined together. Make sure they are in good contact with each other. Hold the iron onto the the combined pieces - as soon as the pieces heat up enough, the solder will melt and all the pieces will effectively become one piece of metal. Remove the soldering iron from contact and put it somewhere safe. The joint will cool very quickly and set hard. You can blow on it to help the cooling. You can burn your fingers if you like by touching it too soon.
If you need to join pieces that cannot be twisted together first, you definitely need to 'tin' them, but twisting together first does away with the need to 'tin'. If this is the case, just heat the joint to be soldered with the soldering iron, applying the solder to the wires to be joined, not the soldering iron.
The soldering iron will take about five minutes to warm up.
There is information and thoughts on soldering irons in the Recommended Equipment section.
Most soldering is pretty similar, but LEDs are a special case. Make sure you know about Soldering LEDs before you try it, otherwise you can seriously reduce their life-span and may even destroy them outright.