I clicked on this link (macOS Mojave 10.14) in the "Download macOS using Safari on your Mac," and when I got to the Mojave download page and clicked "Open," it says, "this copy of the install macOS Mojave application is damaged..." What do I do now?

Upgrade to the latest OS directly. You don't need to go through Mojave.

If it's the latest version compatible with your model or otherwise desired, choose Applications from the Finder's Go menu and throw the existing installer away.

(200230)

If you get a corrupted message...

Naw, not damaged, it's just that you have an old copy already in Applications, there are 2 cures...

Disconnect from the Internet & set the Date back to 2014, then you can run the Install MacsOS Mohave App in Applications, so expired Certificates are not expired.

Or... trash the Install MacsOS Mojave.app in Applications, empty trash & restart, then a new copy of a more recent version can be downloaded.