How should we as disciples understand the Buddha’s wisdom?
Suggested core reading:
Rahula, Walpola, What the Buddha Taught (Second and enlarged edition 1974; reprinted by different publishers)
Further reading:
Choong, Mun-keat, The Fundamental Teachings of Early Buddhism : A Comparative Study Based on the Sūtrāṅga portion of the Pāli Saṃyutta-Nikāya and the Chinese Saṃyuktāgama (Series: Beitrage zur Indologie Band 32; Harrassowitz Verlag, Wiesbaden, 2000).
Might help to shed some light on things … 
"“The fact that when greed is present within you, you discern that greed is present within you; and when greed is not present within you, you discern that greed is not present within you: that is one way in which the Dhamma is visible in the here-&-now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.”
—Anguttara Nikaya 6.47
“Inviting verification” is described in the Buddha’s own path to awakening in Majjhima Nikaya 19, where the results of wholesome and unwholesome thoughts are investigated. In this way the practitioner proves the second & third noble truths in their own experience.
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Wisdom is knowledge that has been discerned to be actual or true and is applied to one’s life in a way in which positively benefits oneself and others.
What makes knowledge wisdom is in the application of that knowledge in a way that is of benefit.
Coming to ‘know’ is a faculty of your own mind. This coming to wakefulness, to knowing, to understanding: is your own capacity for Buddhahood. The difference is that a being advanced in dhamma knows the varying dhammas. The capacity to know resides in all living beings.