Learn Italian: Non ho sottovalutato il mio istinto. - I did not underestimate my instinct.

It is formed by combining the prefix sotto- (under) with the verb valutare (to evaluate or assess). If you ever need to say "to overestimate," you can use the opposite prefix to make sopravvalutare.

If I wanted to drop il mio istinto and just say "I didn't underestimate it," would I use ne?

No, you wouldn't use ne here. The verb sottovalutare takes a direct object, so you would use the direct object pronoun lo, which drops its vowel before the auxiliary verb ho: Non l'ho sottovalutato. The pronoun ne found in this lesson's pronominal verbs (like accorgersene or pentirsene) usually replaces phrases starting with di, which doesn't apply to this verb.

Why do we need the article il before mio istinto?

As a quick refresher: Italian almost always requires a definite article before a possessive adjective and a noun (literally "the my instinct"). You only drop the definite article when talking about singular, unmodified family members, like mio padre or mia madre. Since istinto is not a family member, the il is mandatory.