Repairing a bike inner tube is not only simple and cheap, but it's also mandatory in respect for the environment. Indeed a punctured tube can be repaired many times before being thrown out. A different matter is when the tube is cut or the valve leaks, since it's more difficult repair the tube. Anyway most of the times the leak is caused by a nail or a thorn, and the hole is very tiny.
Although many of you already know this process, I'll explain how to easy add a patch to a pierced bike inner tube.
Step 1: The necessary
All you need is a pump and a repairing kit. I take for granted you already know how to remove the wheel from the bike and the tube from the wheel, so I don't deal this topic here, ask in comments if you fall in some obstacle.
The repairing kit is composed by some patches (probably of different sizes), some scratch tools (sandpaper or a little metal grater), a little tube of mastic (a type of glue) and maybe some little rubber tubes to spread mastic.
Step 2: Inflate
First thing to do: find the hole. This could be simple or pretty long. If you're lucky the hole is big enough to listening the air flowing out when you inflate the tube. But many times the hole is so tiny that air comes out very slow and it's difficult to find, in this case go to next step.
Step 3: Blblblbbl
To find the tiny holes inflate the tube. Don't be scared to inflate the tube very much, it could also become double in dimensions without exploding, and this will help you find the hole, since some hole remain closed with low pressure. At this time dip the tube into water, a sector at a time, until you see the bubbles coming out.