"Trust experts except when you don't"?

Scientists can be wrong. Certain kinds of science are more likely to involve screw-ups. Learn to identify these kinds of findings and learn to identify sources of screw-ups so you don't fall for them.

"Don't trust experts; become one yourself"?

If two experts disagree about something and you want to evaluate the disagreement one way is to understand their arguments. Sometimes you can look into both sides and discover that one of them isn't really the expert you thought they were. You can evaluate the arguments or evaluate the expertise. I can't think of anything else.

Why would I expect myself to be better at judging these kinds of problems than experts who spend their lives on it? Should I not expect myself to be just as bad at it, and potentially much worse (know enough to be dangerous)?

1. I assume you're not planning on trying to publish statistical analyses so I doubt you're dangerous.
2. You can probably learn more about statistics than at least some of the shoddy scientists out there. If you find yourself disagreeing about stats with a prominent statistician then, yeah, you're probably wrong.
3. You aren't learning how to run different kinds of statistical analyses. You're learning about statistical errors scientists make. It's a different set of knowledge which means you can know less about statistics in certain ways but still be able to point out where scientists go wrong.