The solder used should be 60/40, 50/50, or 40/60 with a rosin core. 60/40 is the ratio of tin-to-lead in the solder, so in 60/40 solder there is 60% tin and 40% lead. I think that 60/40 is the best in electronics, but any of the three will do. And rosin is a type of flux (flux is used to clean the metal so the solder will stick), and it is what is used in electronics. But only rosin flux should be used, any other type of flux will corrode and destroy components, component leads, and circuit board traces. The size of the solder is not too important, but you might want to use a thin solder so that you don't apply too much.
Before you solder, you will need to tin the tip of your iron. Tinning the tip is just putting a coat of solder on the tip. This can be done by applying some solder to the tip, and then wiping it off on a damp sponge or rag. The tip of the iron should then be nice and shiny.
After everything is clean you can start to put components in. It is a good idea to put in the smaller components first, to make it easier. Bend the leads of the component if necessary to fit in place. Then put them through the board, and bend the leads to about 45° so they stay in place, and then you can begin to solder it. Take the tip of the iron and touch it to the pad and the lead to heat them both up. Then on the opposite side begin to apply solder. Be careful not to apply too much solder. The solder should spread all over the joint, and make a shiny connection. Remove the solder and then the iron. If the joint was not heated properly the solder will bubble, and make what is called a cold joint. Any cold joints will need to be desoldered, and then soldered again (see How to Desolder).
It is sometimes easier to solder to the PCB when the pads are tinned. To tin the PCB pats, heat the pad with the soldering iron, then apply a small amount of solder, just enough to coat the pad. Then using desoldering braid, remove the excess solder (see How to Desolder).
Heatsinks are good to use when soldering semiconductors and other components that can be easily damaged by heat. Heatsinks are just aluminum clips that absorb some of the heat going to the component, and help prevent damaging the component from too much heat. To use a heatsink clip it onto the lead between the component and joint to be soldered. When soldering ICs you may want to put in a socket, so you wont have to worry about the IC overheating from soldering it in place, and it makes it easy to replace the IC if it ever breaks.
You may wish to cover the joint with heat shrink tubing, to insulate the joint. To use heat shrink tubing, find the smallest tube that can still fit over all the wires. Put the tubing over one side of the wires, solder the joint, and allow it to cool. Then move the tubing over the joint. Using a lighter or match (the soldering iron will not work well) heat the tubing so that it shrinks completely, and tightly covers the joint. When heating the tubing, don't put the flame too close, or the tubing will burn a little, even though they are usually fire resistant.
to insulate the joint, you may want to use heat shrink tubing. To use the heat shrink tubing, first find the smallest tube that can still fit over the terminal and wire(s). Put the tubing over the wire(s), and solder the joint. Allow the joint to cool, then move the tubing over the joint. Using a lighter or match (the soldering iron will not work well) heat the tubing so that it shrinks completely, and tightly covers the joint. When heating the tubing, don't put the flame too close, or the tubing will burn a little, even though they are usually fire resistant.