How do we register a Mac warranty?

During the initial setup process you are prompted to log onto the Internet. If you did that you automatically registered your computer with Apple and your 1 year warranty (longer in some parts of the world) began on the purchase date. If you purchase an AppleCare extended warranty the package has instructions on how/where to go online to register the computer with its extended warranty and if you purchased AppleCare and the computer from Apple at the same time AppleCare is automatically registered, there is nothing extra to do.

You can verify if your device is registered with Apple by entering its serial number into this Apple support page:

https://checkcoverage.apple.com/

I bought a new iMac only two weeks ago and did exactly the steps that dwb lists--nothing more. Using the above tool confirms that Apple received the information without my having to jump through hoops: