I would recommend learning about skepticism, but not the kind you will find in the USA to be honest, the USA skepticism community has some bad actors mixed in and it is hard to get the good without the bad.

Skeptics with a K is a podcast from the Merseyside Skeptics Society and they do great work. If you start there and listen through a little they will recommend other podcasts and websites and you can go from there.

The reason I suggest them as a starting point is their attitude. They are compassionate people who see the harms of woo and religion but really so embody kindness and connection over anger and hate. They also embrace the kind of skepticism where you maintain a focus on the tools but don't get locked into extreme positions, exactly what you are talking about. They practice not running to the extremes, but not by saying things like "both sides are the same". They have convictions, but they are slow to build and carefully constructed. If you copied their mental processes you would be in a fairly good starting position.