Let's say first you want to measure the speed of light. One way would be to send a light beam out to some object and see how long it takes to get back. Since the speed light is very fast, you need to have two objects that are a known fixed distance from each other and are far enough apart so that you can measure the time difference between the signal you send and the signal you receive back.

A scientist named Albert Michelson came up with a very clever way to do this measurement. I'll leave it up to you to find out what he did. The info should be in any basic physics textbook. There is also some info. on the Internet although I couldn't find a really good description. Here's some pages to look at:

http://hum.amu.edu.pl/~zbzw/ph/sci/aam.htm
http://www.phys.virginia.edu/classes/109N/lectures/spedlite.html

Now, let's say you get in a spaceship and travel very close to the speed of light for a while and then stop. Since, according to Einstein's theory of special relativity, you can never go faster than the speed of light, then you also won't be able to look back and see yourself grow up; the light is always "ahead" of you.