I'm not even certain I know what is meant by that phrase, "critical thinking."

I've been asking myself that for years. I think I'm getting better at it, but I still have difficulty.

#1: Recognize faith (belief without evidence) as being devoid of any truth (demonstrable and testable) merit. This will likely expose religions as the childish world of make-believe superstition.

#2: Carefully analyze situations and relationships based on facts (testable, things that can be demonstrated) i.e. form world views based on reality, not fantasy.

#3: Question all the sources for your information.

#4: Recognize your (man kinds) bias and shortcomings in accurately analyzing information.

I would highly recommend watching many of the videos from the youtube user anticitizenx.

Spend some time and check out [atheist-experience.com] There are many examples hidden in those archives that demonstrate examples of critical thinking skills.

By being open to the idea that we may be wrong, and having the ability to change our mind when presented with compelling evidence that contradicts previously cherished beliefs. Not rushing to judgement, but considering all evidence and arguments, pro and con, before reaching a conclusion. Having the courage to go against the herd although it seems you may be in a minority of one. Reading and researching thoroughly from unbiased news sources and taking account of counter arguments prior to forming a belief.

It may just be that we need to LEARN not to simply accept but instead seek the answers for ourselves.

Dr Steven Novella has a course on critical thinking. You can find it on “The Great Courses”.
He also has a podcast “The Skeptics Guide to the Universe “. He has written a couple of books on science, evidence based and critical thinking.
Sorry if it sounds like an ad but his podcast was the first I found on atheism and critical thinking and it helped a lot.

Simple, but not easy. We must master the techniques and craft of critical thinking, and be presented with live models of the process by people in public life. There is simply too damned little of it.

Questioning everything is a start. Who, what, when, where, why and how...and if it doesn't make sense, keep questioning until it does or proves to be false

Reading this right now - it is a wonderful primer on critical thinking and logical fallacies [amazon.com]

In addition to what was already mentioned, teaching and learning logic and grammar would be a great start. If you are wondering how grammar would help one become better at critical thinking, I suggest getting the book, "Writing and Thinking", by Foerster and Steadman. The authors point out that writing and thinking are linked. “If words are dependent on thoughts, thoughts are also dependent on words.”

First we have to become aware of our own biases. We have to be humble and don't overestimate our abilities. We can fool others but we should never fool ourselves. We should try to disprove the believes we hold to a much greater extent than we criticize the believes of others. Also we should practice analyzing arguments in order to learn to tell a valid from a fallacious one.

According to Mr Alcock in the Skeptical Inquirer the following points can aid a person. Remember everyone can be fooled.Be cautious of your personal intuitions. Be wary of Fundamental Attribution Error. Be wary of validation by personal experience. Use more than one source of information. Do not over interpret correlations. When inadequate evidence exists suspend judgment Finally ask compared to what? These points have kept me out of problems when confronted with new ideas, different political positions and becoming an atheist verses following a religion.