How do I become a Werewolf?

Raven Dreamer's answer doesn't cover the second part of the question:

And by extension, how do I cure lycanthropy, that is, if there even is a cure?

Fortunately, you are also introduced to the cure when you are doing the Companions quest line. You'll be asked to kill some Glenmoril Witches by Kodlak, and make sure to hang onto their heads. The questline will tell you exactly how to cure your Lycanthropy.

The upper Echelon of The Companions (Fighter's Guild), called The Circle, is composed of Werewolves. As you advance through the Companions quest line, you will eventually be inducted into The Circle, whereupon they will perform a ritual to turn you into a werewolf.

(You need to become a werewolf to continue the Companions questline, but you aren't forced to complete the quest (and become a werewolf) if you don't want to).

And by extension, how do I cure lycanthropy, that is, if there even is a cure?

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Fortunately, you are also introduced to the cure when you are doing the Companions quest line. You'll be asked to kill some Glenmoril Witches by Kodlak, and make sure to hang onto their heads. The questline will tell you exactly how to cure your Lycanthropy.

Source: Cthos