I'm a Christian now, but I can remember the days when I was a rather unreflective atheist. Back then, I considered the word "wisdom" to be meaningless, and would be suspicious of anybody who claimed to have it.
Now, as a Christian, I think of myself as having a certain degree of wisdom, and consider it to begin with "The fear of the Lord". But what does it mean for the reflective secular person? How is wisdom obtained within the secular or atheistic worldview?
Wisdom can have several different meanings. Aristotle in his practical philosophy distinguishes three Greek words that reflect "knowledge": (1) episteme, (2) techne, and (3) phronesis.
episteme refers to the sort of knowledge of facts that we often call knowledge in English.
techne refers to what we sometimes call "technical skill." It reflects the ability to accomplish something and produce a product. The classic example is a skilled blacksmith or carpenter.
phronesis is often translated as "practical wisdom" and it reflects the skill in making wise choices. For Aristotle, it's a necessary foundation for all of the other virtues.
I wouldn't say Aristotle's distinction is perfect, but there's something to be said for pointing out the difference between being say an excellent forester and the world's leading tree botanist with the former being "wiser" and the latter being more "knowledgeable".
At least as far as I read Christian Scripture, the sort of wisdom generally referred to there seems like it is either something like phronesis or an idiosyncratic usage that means "not being a fool who doesn't acknowledge God exists or believes in him."
The two senses may not be infinitely far away. You might want to look at Linda Zagzebski's book on the virtues and the role of practical wisdom in Aristotle and Aquinas for more.
In terms of your question as to how one obtains wisdom outside of a Christian account, the answer will hinge on whether you are thinking of wisdom as practically living well or as believing in God (or how you think the two definitions interact). The more strongly you believe it's the believing in God side the less philosophical your question is. The more it's about living well, then the more Aristotle's answer will seem to fit.
For Aristotle, the answer is that you need to be raised well (Nicomachean Ethics BK I), learn how to master your desires such that you don't confuse pleasure for good (Nicomachean Ethics BK II), cultivate in yourself virtues, and engage in a life of the mind. (I'm skipping over quite a bit).
Why not ask a dictionary?
a : accumulated philosophic or scientific learning : knowledge
b : ability to discern inner qualities and relationships : insight
c : good sense : judgment
d : generally accepted belief (challenges what has become accepted wisdom among many historians — Robert Darnton)
That sounds about right. The definition is far more general than any particular belief system; as is typical with colloquial language, "wisdom" is not precisely defined but is some blend of the four points above such that it seems sufficiently distinct from other things we recognize and give names to.
If you get too attached to using "wisdom" to mean "particular insights regarding my non-universally held beliefs", you're liable to end up with conversations like this:
"Fear of an imaginary creature postulated to be supremely loving is wisdom?!?!"
"Denying the existence of the creator of everything and the source of all knowledge and value is wisdom?!?!
I can't see that using language this way would be particularly helpful.
(As an aside, there are plenty of highly reflective atheists, so take care not to muddle issues of faith with issues of reflection.)
Edit: I didn't actually say how an atheist would do these things because it's not any different from how anyone would do these things (apart from e.g. praying and Bible study). To get knowledge, study; to get insight, try to discern and test causes and effects; to get judgment, note carefully the consequences of different actions. Reading great literature, or about current events or history will give you plenty of material from which to glean insight and judgment.
Some good points raised in the other answers, but I think there was one thing missed: many Christians when they study the ideas about the virtue of wisdom (also known as prudence) look to Augustine and Aquinas. Augustine and Aquinas however, looked to Cicero and Aristotle respectively. The point then is that the first source for Christian ideas were non-Christian philosophers!
A Catholic priest named Robert Barron gave a talk that I cannot find online anymore, but the talk was an overview of all seven virtues. Two points he made that are relevant to the question:
Christian wisdom and secular wisdom can usefully be compared. In their difference they are mutually elucidatory. The (secular) philosopher, John Kekes, is helpful here :
There are two traditions of thinking about [wisdom]. One claims the allegiance of Plato, Aquinas, Descartes, and their disciples. The other has the support of Protagoras and Montaigne, and it has achieved its first full formulation by the insufficiently appreciated Charron in De la Sagesse.
Augustine's rethinking of the meaning of wisdom illustrates [the two traditions] .... [H]e alters its meaning by splitting it into two parts, distinguishing the knowledge of divine things from human things and limiting the object of wisdom to the divine. Knowledge of human things is scientia. Wisdom is ... sapientia ... an intellectual cognition of eternal things.
Both traditions think of the knowledge involved in wisdom as providing a perspective upon the human situation. For the first, metaphysical, tradition this perspective is sub specie aeternitatis; it views human affairs from the perspective of eternal things, God, or the unchanging rational order of reality. For the second, humanistic, tradition the perspective is sub specie humanitatis; a view of the human situation from the human vantage point. For the first, the possession of wisdom consists in understanding and willing the rational order of reality. For the second, wisdom is to arrange human affairs for the benefit of humanity in the midst of an indifferent reality. The distinction between the two traditions is not absolute. The metaphysical view is compatible with love of humanity; nevertheless, this love is secondary, and it must be informed by knowledge of eternal things. Similarly, the humanistic view is not indifferent to impersonal knowledge of reality; indeed, science is one of the glories of this tradition. All the same, it insists that the human point of view is unavoidable for humans. It should not go so far as the mistaken Protagorean relativism believing that man is the measure of all things, but it should recognize that man is the measurer of all things.
My understanding of wisdom is humanistic. The theoretical aspect of it is knowledge sub specie humanitatis. And the objects of this knowledge ... "are not the specialities of the most refined thinking ... [but] ... the commonplaces of the least refined thinking." More specifically, it is interpretive knowledge of the human significance the universal and unavoidable limitations and possibilities have for living a good life. How, then, is wisdom, thus understood, action-guiding?
It is action-guiding negatively: by issuing warnings of what not to do if a person wants to have a good life. Wisdom is corrective. It reminds the unwise of the relevance of their own descriptive knowledge to their pursuits. The proper occasions for such reminders are those frequent lapses in which we are attracted by ideals whose achievement is impossible due to human or personal limitations. Wisdom guides action in two ways: by differentiating between what is possible and impossible for anyone, and by drawing the same distinction for a particular person in his context. Wisdom licences the possible and warns against the impossible. It identifies both the ideals to which a person may reasonably commit himself in pursuit of a good life and the ideals incompatible with living well. (John Kekes, 'Wisdom', American Philosophical Quarterly, Vol. 20, No. 3 (Jul., 1983), pp. 277-286 : 281-2.)
References
John Kekes, 'Wisdom', American Philosophical Quarterly, Vol. 20, No. 3 (Jul., 1983), pp. 277-286.
E. F. Rice, Jr., The Renaissance Idea of Wisdom (Cambridge: Harvard, 1958), p.4.. [Study of Charron and his tradition. Book is still illuminating.)
D. Collins, The Lure of Wisdom, The Aquinas Lectures, 1962 (Milwaukee: Marquette University Press, 1962). [See esp. discussion of Descartes.]
