Racism, sometimes it's very subtle.
On thinking for yourself:
Ask questions, particularly the question "why?". Ask everyone (not just the so-called experts), and try to answer your own questions as well. When you get an answer, try to think of exceptions, and then ask yourself why those exceptions exist. Never be satisfied until you arrive at an answer that has very few exceptions.
Look for selfish motives. Some people will become very annoyed, and perhaps even offended, that you're questioning something they accept without question. Whenever people want you to think a certain way, it's because it benefits them in some way. But that benefit is not always obvious or direct. Many times, people want you to adopt their perspective because it makes them feel more comfortable and secure (safety in numbers). Sometimes, people's beliefs make it easier for them to feel like a good person. These people don't want those beliefs challenged because it's as if you were challenging them personally - it seems to them that you are questioning their "good-person-hood". Sometimes, people are trying to look out for your best interest, and truly want you to be in step with their beliefs without looking into their statements any further. And sometimes, people just want to be seen as authoritative and trusted, so they're personally invested in whether or not you buy into the things they say. That's why they take it personally if you don't automatically buy in.
Stop being a people pleaser. People who don't think for themselves are often scared of disagreeing with others, and scared of "rocking the boat". A freethinker, on the other hand, bases their self worth on something other than what people think of them. These people may still experience rejection, discomfort, and anguish, but they will continue to think for themselves.
In cases where someone says he "just wants the best for you," you may be accused of distrust, and it could make you feel guilty. But keep in mind that anyone who truly cares for you will be willing to explain their point of view and why they feel that way, and allow you to decide for yourself whether that is enough evidence for you.
Do the research. Look into the statements made by others. You'll be amazed at how many times you'll find lots of evidence to contradict the statements of others. Yet, these people spout this erroneous information and never question the accuracy or truth of what they're saying. Use Google or go to the library, and search for information to prove or disprove the statements made. Remember where you get the "evidence" from. Be aware that, just because you saw it in a book or on the internet, that alone does not make it the truth. Once you've found evidence, one way or the other, you can speak up about it. "Yes, you know after we talked last time, I was so interested that I looked that up. That's amazing, isn't it, hard to believe, but true!" Or conversely, you can say, "I know that sounds amazing, and I hate to burst the bubble because it's fun to believe that could be true, but I looked it up, and it looks like it isn't true. I feel bad to be the bearer of bad news, but I just don't think that's true. You can look at ____________ (wherever you found your disproving evidence) and see for yourself."
Be humble. When you're breaking the news that your friend is passing along a false tale, let them know in a humble and compassionate way - don't just come in crowing and congratulating yourself for debunking a myth. You may look smart to others for a minute, but to your friend, you look like a jerk.
Live outside your comfort zone. Not only will some people be very perturbed by your refusal to take their statements at face value, but you will also learn to question your own assumptions, and that can make you feel lost and confused, like walking into a dark room. It takes courage to face uncertainty.
Be Bold.
Beware paralysis by analysis. When you're thinking for yourself, you're taking full responsibility for your life and your actions, because you can't say you were trusting someone else's judgment. This can be very nerve-wrecking, and lead to excessive self-doubt. Remember that thinking for yourself doesn't mean being sure. It means making decisions based on your own analysis, rather than someone else's. There will always be some degree of uncertainty, no matter what, that you must learn to accept and cope with.