I don't even think in those terms i.e. establishing trust with a source, as even the most trustworthy sources sometimes falter. Instead I'll try to verity the information, and if the information coming from some source repeatedly proves to be biased and misleading then that source is likely not very credible.
Academia/professional field: typically within any field there is a journal or several journals that have are both peer reviewed and have a high impact factor - these are the ones you want to subscribe to and occasionally read to keep track of what's happening in your professional field. Peer reviewed means they aren't publishing any random article they received and high impact usually holds them to a least some standards in publication. If you do not know what these are, you can ask your coworkers or your boss, and check out the references on articles to see which sources are available out there.
There is something that usually for both academia and personal situations and that is
concilience - this is when you see multiple independent sources corroborating the same information, so for example 3+ research groups from different universities finding the same kind of results, or on personal level 3+ different people telling your the same thing about your work. This increases the likelihood that there is something objective there and they aren't presenting you with skewed and subjective interpretations.
News/politics: try to figure out their group affiliations and biases; read news from multiple sources, both left and right, your home country and other countries, this way you get well-rounded information that isn't slanted to support only one group. After participating in both left and right leaning groups in US I would say that politically oriented information has the highest probability of getting distorted and faked, because they are just out to sway opinions most of the time.
On result sensing: who/which group actually produces results that are of value and interest? they are likely to have something figured out that is reliable.