In addition to what leaf mentioned, and the classic Google test, you can also use NameCheckr (https://www.namecheckr.com/) to quickly and efficiently see what domains/handles/usernames you’ll be able to land-grab with your chosen name.
Trademarking definitely only worth exploring once you've shipped, and if the game becomes successful. This is not something you need to worry about or budget for at an indie/solo-developer level.