Is there a word referring to the process of digesting information and coming up with your own opinion/ideas on a topic?

It could be used in a sentence like "With such influences in the media leading people to believe everything they hear without ... "

The word synthesis fits the definition in the question title quite well.

synthesis

1 c. : the combining of often diverse conceptions into a coherent whole

2 a. : deductive reasoning
b : the dialectic combination of thesis and antithesis into a higher stage of truth

https://www.merriam-webster.com/dictionary/synthesis

But that word wouldn't fit your example sentence especially well. It's too technical, scholastic, or extravagant for what is a very common activity (or what should be very common).

Thinking, as suggested in a comment is a reasonable choice. I'd prefer critical thinking, which doesn't have a substantially different meaning, but is used frequently in this context.

Another similar choice that fits the example sentence better than the definition in your title is reflection

reflection

6 : a thought, idea, or opinion formed or a remark made as a result of meditation

7 : consideration of some subject matter, idea, or purpose

https://www.merriam-webster.com/dictionary/reflection

How about ponder?

To ponder is:

1) to weigh in the mind: Appraise or

2) to think about: reflect on

That would fit with your sample sentence.

Then there is deliberate, which means:

to think about and discuss issues or situations carefully

This would be a less likely fit. But another good one just came to mind:

evaluate

which can mean:

to form a critical opinion of

You could fill in your sentence by saying: "With such influences in the media leading people to believe everything they hear without evaluating the information "...