It may be possible to file for a trademark for the slogan as long as it's used to indicate & identify the source of goods/services.

Find out more on how to trademark a slogan

Find out more on how to trademark a domain name

You wouldn't copyright it. Copyrights can be obtained for things of an artistic nature. This includes, of course, poetry, films, sculptures, music, fiction, etc.

To protect a business name within your industry, you would apply for a trademark.

Find out more on how to trademark a business name