Just go to radioshack and pick up a 40watt iron and some rosin core solder. Rosin core has flux in the solder, that's what draws the solder to the copper wire. You're only joining wires, the type of solder you use isn't very important. I think 60/40 is what radioshack generally stocks and it works fine. I like to use the thinner solder, it's easier to control how much solder you use. Using too much solder can leave a sharp solder joint that can break through heat shrink tubing. Grab so me flux paste too, if you run into any wires that are slightly corroded or heavier gauge wire extra flux will help it stick.
I personally do not tin the wires, only the tip of the iron. Tinned wires are stiff, and you can't twist them together. It's hard to explain but you wanna twist the wires together, but not the way you would if you were using a wirenut, like this "- - " not like this "_|_". Remember, heat the wires with the tip of the iron and let the solder melt into the wire. Don't melt the solder with the tip itself.