A trademark registration for each design iteration of a logo can be expensive. However, it is not necessary to register each individual design to gain protection. Register the logo as a "standard character mark," rather than a design mark. By registering the plain name rather than the design, the mark is protected in all its iterations, regardless of colors, embellishments or typestyles used.see Words vs Logos by attorney Erik M. Pelton
Because of this versatility, it is more common to register a logo as a standard character mark rather than as a design. If you register only the design, then once the design changes, you will need to register the new design.
Some companies, however, choose to register a logo as a design in addition to registering it as a standard character mark. They may even apply for three registrations for a logo: one as a standard character mark, one as a design mark in color, and a third as a black & white design mark. This, presumably, ensures the ultimate level of protection for a logo that has high value (the Coca-Cola logo is said to be worth 70 billion dollars).
Trademark registration typically takes about a year if there are no snags. If the US Patent and Trademark Office generates an "office action" in response to the application, it could take longer. Some cases have taken years.
Regarding copyright protection, many logos do not qualify for it. The US Copyright Office, as well as the US congress, has had a long-standing policy that designs consisting merely of lettering, even if embellished, and simple shapes, are not protectable by copyright. This would disqualify a vast number of typical company logo designs for copyright protection.
US Copyright FAQS
If, on the other hand, T-shirts are printed with original artwork or photographs, they are probably protected by copyright. Copyright protection begins automatically the moment a work is created in "fixed" form. Printing to a shirt is obviously a fixed form. A photograph is a fixed form. Even saving a work to a hard drive is a fixed form.
Copyright registration is fairly straightforward and inexpensive and can be done online. Trademark registration is far more complicated and is not cheap.
Copyright protection lasts for a fixed time. It typically lasts for the life of the author plus 70 years. A copyright created by a corporate entity lasts a flat 120 years.
A trademark registration can, in theory, last forever as long as it is maintained.
Costs for registration, penalties for infringement, and the standards for determining infringement vary significantly between copyright and trademark protection.
A wise first move for anyone wanting to protect a logo or original work would be a consult with an intellectual property attorney. Often these initial consults are inexpensive.
Erik M. Pelton, referenced above, is both an intellectual propertty lawyer and a former examiner for the US Patent & Trademark Office. His blogs and videos contain much valuable information for designers. And they are written in plain English, so even I can understand them.