Are you in the United States? Your business name is protected just by you using it, at least in proximity to where you are. For nationwide protection, a trademark registration would cost you around $400 for each category of goods or services you want to register the name for. Is that - given your budget and financial plans - very expensive? I'd wait then. As you have registered the internet domain name as well as the social media accounts and your business is online anyways, I'd say start and see how it develops.
Would you like to explore the opportunities of strengthening your brand and its values right from the start of your company? Consider to schedule a call with me; in just 15 minutes we can already achieve a lot!
Good luck with the new company!
Rogier