Learn Russian: Мне не нравятся насекомые!

Насекомые is a substantivized adjective—a word that functions as a noun but declines exactly like an adjective. Historically, it meant the notched ones (referring to their segmented bodies), but today it is simply the standard noun for insects.

Since насекомые is plural, how do I say a single insect?

You use the neuter singular adjective ending: насекомое. Because it acts like an adjective, it will decline through the cases exactly like a regular neuter adjective would (for example, одно маленькое насекомое for one small insect).

Why do we use the plural verb нравятся here instead of нравится?

With the verb нравиться (to be pleasing), the thing you like is actually the grammatical subject of the sentence. Since насекомые (insects) is plural, the verb must also be plural (нравятся). Literally, this sentence translates to: Insects are not pleasing to me.

Just a quick refresher: why does the sentence start with Мне instead of Я?

Because the person experiencing the feeling in sentences with нравиться must be in the dative case. The insects are doing the pleasing to me, so я becomes мне.