In terms of a logo, copyright only extends to ownership of the original design/artwork. Your copyright protection ceases when you “industrially” apply your logo to your goods and/or services.
When applying to register a trade mark you ought to ensure that your logo does not infringe copyright. If a designer has created your logo for you, you need to ensure that any copyright logo has been assigned to you in writing as payment alone does not automatically constitute a transfer of copyright.
You also cannot register a trade mark which contains stock images or illustrations. Fonts/typefaces are also subject to copyright protection, so it is a good idea to ensure that your license grants permission to incorporate the specified font in a commercial trade mark.