The main feature of truth-seeking, the search for truth, or being a truth-seeker, is the attempt to discover deep truths, especially by putting aside simple prejudices and unexamined traditions. The notion of truth-seeking, or the search after truth, has a very long history, associated especially with philosophy and religion. While these terms used in many ways, it often means not merely a casual practice but a whole way of life, one that is devoted to discovering previously-hidden truths--so that the academic or scientific life is sometimes called a life of truth-seeking.
The more modern phrase "fact-finding mission" does not have the same ring as "truth-seeking." A fact-finding mission tends to be done by a group, and to result in a report of alleged facts that a group can agree on. Truth-seeking, by contrast, is largely a solitary affair, because the doubting it involves, and the embrace of "higher" truths it might elicit, are solitary acts.
It is probably more often used to describe a way of life of a certain kind of philosopher or religious devotee, one in which the truth-seeker has conscientiously set aside as many of his prejudices as possible. This is typically described as a difficult and personally wrenching experience, one that many may feign by pretending to be jaded and skeptical. Yet the truth-seeker often appears not at all jaded but, instead, very earnest, serious, and perhaps naive. In this sense, Socrates and Descartes were truth-seekers par excellence--and, not surprisingly, both were associated with skepticism and had some well-developed ideas about the possibility and conditions of knowledge. They disagreed, however, in that Socrates claimed not to have found the truth with any certainty, while Descartes claimed the opposite. It was Descartes who wrote an incomplete, unpublished dialogue titled The Search for Truth by means of the Natural Light (of reason).
In a Christian context, "the search for truth" is sometimes understood to mean the search to know some higher, divine truth, especially the logos (word, truth) that John 1:1 says is Jesus Christ himself.