Do I apply for trademark protection or copyright protection? And how many do I apply for?

This is in the domain of trademarks. You should register each and every dog design you intend to use in your business. Registration with the USPTO is very good because it gives you nationwide rights to the designs in question. If you want to read up on trademark law, go look around on the USPTO website, or google the Lanham Act. It may be worth your while to have an attorney help you in the process of filing for a trademark.