This is a broad question. How justified should your belief in a claim be? Your belief should be in proportion to the evidence. Here are some heuristics for evaluating the evidence given certain types of claims.
For breaking news, I find On the Media's "Breaking News Consumer Handbook" useful.
For fact-based claims (did X happen?, did X say Y?, etc.), following the standards of good journalism will help lead you to the correct answer, or at least avoid incorrect answers. NPR's ethics handbook gives a set of tactics. Here are just a few:
- seek diverse perspectives
- double-check numbers and the way they're portrayed
- be skeptical of images; start with the presumption that they are not authentic
- identify sources for each fact you are relying on
For more nuanced claims that involve scientific uncertainty (e.g. does X cause Y?), this is a difficult task to do right. You have to familiarize yourself with the large body of research on a topic. Give more weight to more rigorous methods and complete reviews. Follow the evidence, and be willing to accept uncertainty in your conclusion. As in all exercises of skepticism, even when think you've arrived at a well supported conclusion, you need to be open to change if presented with new evidence.
Be on guard against pseudoscience. TEDx/TED distributed a letter to TEDx organizers with heuristics to help them identify pseudoscience. Here are a few:
- A mark of good science: It makes claims that can be tested and verified
- A mark of bad science: Is not based on experiments that can be reproduced by others
- Red flag behaviour: Provides data that takes the form of anecdotes, testimonials and/or studies of only one person