You can have trademark rights without registering your trademark. Those unregistered rights are sometimes referred to a common law trademark rights.
You have trademark rights upon using the name, such as by placing the name on your product or by placing your name on your website if you offer services
You can have trademark rights separate from registration
Apart for doing a Google search, you might use one of the many vendors that offer trademark searches. These searches can include common law trademarks (i.e., un-registered trademarks).