When soldering wires together a systematic process makes it a lot easier to learn. Having the correct equipment to begin with will make it more likely to be a success.
Stripping the Wires
Using a pair of wire strippers, remove about 10mm of each of the end of wire that you want to solder together.
Splicing the Wires
Mount the wires in a set of helping hands.
Preparing for solder
Twist the wires together to make a secure connection between them insuring the all of the strands are twisted and none are poking out.
Apply a small amount of flux to the two wires.
Applyling the Solder
Hold the soldeing iron onto the wires where they join and slowly feed in some solder into the joint. Quickly remove the end of the solder real followed by the soldering iron.
Cleaning the Solder Joint
Using some flux cleaner and a brush. Apply some flux cleaner to the solder joint and brush until any flux residue has been been cleaned away.