The Scientific Method, as I see it, can be divided into two parts, imagination and skepticism. You use imagination to develop new ideas and apply skepticism to test the merits of the ideas. These twin engines of creativity and testing have powered scientific and technological breakthroughs throughout history.
If you want to know more about skepticism, here are some books I recommend.
The unofficial "bible" of skepticism, and one of the greatest books I have ever read, is The Demon-Haunted World by Carl Sagan. Sagan not only goes over the basics of critical thinking, but also describes why critical thinking is important. The chapter called "The Dragon in my Garage" is one of the most well-known refutations of "goalpost-moving" tactics like the Omphalos argument. Briefly, if there is no difference between an incorporeal dragon and no dragon at all, why should we accept the existence of the dragon?
While Sagan's work covers why we should be thinking critically, Michael Shermer's Why People Believe Weird Things, explains why people often don't think critically. One of the biggest reasons is that human brains seem to be wired for "pattern-seeking" and not wired for testing those patterns they find (confirmation bias). Shermer explains how people come to believe different things and why they continue to believe those things even after proven false.
Next up is a personal favorite, though it isn't as mentioned as much in other places. John Allen Paulos's Innumeracy examines the perils of being a math-illiterate society and individual. While the average person knows a little algebra, their grasp of probability and coincidences is often weak. Con-artists, pseudoscientists and advocates of various causes all use bad math, intentionally or accidentally, to promote their products or philosophies. If your understanding of mathematics is good enough, you can often see through the falsehoods.
For specific applications of critical thinking, I recommend Lynne Kelly's The Skeptic's Guide to the Paranormal. Kelly covers most of the basic paranormal and unusual claims by going over the various arguments for a claim and the skeptical refutations. There are a lot of interesting science tidbits used to dispel several of these claims, you'll probably learn something new as well.
Finally, if you want to see critical thinking applied to the myths and misconceptions that hang around the sciences, you can't go wrong with Phil Plait's Bad Astronomy. In it, Phil debunks moon landing hoaxes, face on Mars claims, global destruction at the hand of giant astroids and standing eggs up on the equinox. You should, if you aren't already, also check out his Bad Astronomy Blog.
So if you haven't read those yet, you've got a lot of pages ahead of you. Happy reading!