Many SaaS Startups are pretty good at getting their first website up, register domain names and picking a logo. They often forget to check the Trademark database though, and this can lead to problems later.
This video is a great intro to what you need to know. If anything, just search the Trademark Database by the US Patent and Trademark Office. It will tell you who else might already be using your planned company or product name.
Check out more on their channel to lear useful tips that will protect you down the line.