If I want to design a logo for a client, am I required to copyright that logo?
Copyright protection is automatic for all original works as soon as they're created. If providing an original design for a client to use as their logo, you usually would need to fully transfer copyright to them. That way the work is no longer yours, and they get to make whatever copies they want. It's possible for you to not transfer copyright and just providing a very generous license to them, but most places are going to want the copyright fully transferred to them so you can no longer mess with their use of the work. When they use it as a logo, they then establish its use for trademark purposes. There's no trademark issue until they start using the work with consumers to establish the source of goods and services. Anyway, you're not required to copyright the logo. It's automatically copyright protected. But you almost certainly would need to transfer copyright to them as part of the deal. Then they'd register it as a trademark.
Pascal the Gambler
You trademark a logo, not copyright, and that would be them, not you. They would own the logo, as you created it under contract for them.
Copyright is INHERENT in anything the CAN be protected by copyright law. A logo would fall under TRADEMARK law, not copyright. That said, any logo created FOR a client would belong to the client, not to you. It would be their job to file for trademark protection unless they contract specifically states otherwise.
The logo will be copyright as soon as you design it. You do not need to take any further action. Your agreement with the client should specify who owns the copyright. Without an agreement the copyright belongs to you, and the client needs your permission to use it (in theory). That is silly, and logically the copyright should be transferred to the client. If the client intends to use the logo as a trademark that requires registration. That is the client's problem, not yours.