How do I appear offline on Discord so I can lurk without people messaging me?

Click your profile picture in the bottom-left corner and you'll get four status options. Pick Invisible. That shows you as offline to everyone while you can still read, type and use Discord completely normally. People can still message you, you'll get it, they just can't tell you're online. There's also Do Not Disturb if you'd rather look online but silence every notification ping instead.

Invisible is exactly what you want. Full access, looks offline, nobody knows. The only tell is if you're typing in a server, so just don't type and you're a ghost.

Worth knowing the difference: Invisible = looks offline but you get everything. Do Not Disturb = shows online but mutes pings. Pick Invisible for pure lurking.