You can pick up a cheap soldering iron at Wal-Mart or Sears for around $15. Get that, and some rosin core solder and you're all set. It helps if you get some flux paste as well. Flux helps with heat distribution, and it cleans the material you're soldering as it is heated.

The process is simple. Plug the soldering iron in, let it heat up. Touch it to the conductor that you're going to solder, and then touch the tip of the solder to the iron and the conductor. The solder will melt and bond to the conductor (wire, switch contact, whatever...). A good solder joint should look fairly shiny and clean. If it is a dull grey then you have a cold solder joint (which means there is a greater chance of the connection failing). If it has a brownish coating that's just excess flux on the joint. You can clean that off with a cotton swab and some rubbing alcohol. Hope that helps.