The first item you will need is a soldering iron itself. It's not a magical device, it is basically just a hot stick of metal with a point on one end and a handle on then other. There are a few different types, but all you will need is a basic soldering iron from Dick Smith or Jaycar. Jaycar Cat TS-1466 for $20 or I recomend the kit Cat TS-1650 for $32 which comes with a stand to hold the iron and a cleaning pad.
Next thing you'll require is some solder. Solder looks like a grey piece of wire. It is made of a mixture (alloy) of Lead and Tin so that it melts at a very low temperature. It also has a chemical indside the wire called 'flux' that acts a bit like acid and helps the molten solder stick to other metals easier. Buy your solder from only places like Dick Smith and Jaycar cat NS-3013 $1.65 because plumber's solder etc that you can buy from a hardware store are slightly different.
That's all you need. You might like some other tools like solder wick or a solder sucker. You will find these things in the various shops if you think you will need them. There is one thing which I do recomend for those soldering tiny components or heat sensitive devices. Dick Smith sell a set of "heat sink clips". These are small aluminium clips that absorb some heat and help protect a component. The are not essential, but they are helpful for newbies.
A book that I highly recomend is the Dick Smith Funway 2 (green book). It has a very good section on how to solder. You can also buy a kit which you will get the book, a soldering iron and two kits to build all in the one package. It's an excellent xmas gift !
If you have the Jaycar kit, and the solder. Plug the iron in and let it heat up for 5 minutes. Yes it gets hot enough to burn so make sure you know which end to hold ! The sponge in the bottom of the stand is for cleaning the tip. Soak it in some water. Unroll some of the solder from it's reel, you can break a piece off if you like, it is very soft.
Ok, if it is hot enough you should be able to touch the tip of the soldering iron with the end of the solder wire and it will melt. There may be a little bit of smoke, but this is normal. Avoid breathing the smoke, consider having a small fan to blow it away.
Unroll some of the solder wick or if you have a piece of circuit board, put the tip of the iron onto the copper. Give it a couple of seconds to transfer some heat. Now touch where the soldering iron and copper meet with the end of the solder wire. The melted solder will flow onto the copper like water. Take the iron off and let the copper cool. Wipe your soldering iron tip on the wet sponge to take away any excess solder.
That's it, the copper should have cooled enough to touch now... you have done you're first bit of soldering.
Now to join two pieces together. Quite often it can be difficult to heat two pieces of metal to solder together... example two copper wires. So we do something called "tin" the wire. It is just a matter of putting some solder on each of the wires first and allowing them to cool. Then, when we hold them together and put the soldering iron on them, the solder on each one will already be stuck to each and it will melt very quickly allowing you to join them easily.
If you have a piece of wire, strip both ends. Now melt some solder onto each end so that it "takes" (melts and sticks) to the wire. Now bend the wire around so that both ends are touching and heat both together with the soldering iron. Add a little bit more solder if required. When you can see that the solder on both ends is melted and joined, remove the iron but hold the wires in place until the solder has set (about 10 seconds). It is done ! you have now soldered two wires together !
If you have a kit that you want to build, then start with a cheap kit first (like the dick smith funway 2 flasher) so you don't wreck an expensive kit. Resistors are cheap to replace and they are very tollerant of heat, they are the first thing you should practice soldering on a PCB. If you have heat sink clips, then attach them to the component legs on the component side of the board to protect it while you solder.
Basically if you have 'tinned' the component leg first, you will touch the soldering iron onto the PCB track and the wire leg at the same time to heat them both, then add a small amount of solder and remove the iron. If you have done it correctly, the hole will be completely covered by solder.
That's basically it ! It's that simple, anybody can do it. If you have a friend who can already solder then you might like to have them show you or give you some help.