It's worth noting that if you make a new account, it seems Threads will instantly suspend you - don't worry though, if you run through a brief verification process, you can appeal the suspension and will (hopefully) get full access in a day or two.
If you do already have an account, your Instagram handle will already be 'reserved' for you to claim in Threads! Otherwise, you can set your username upon creating an account. Instead, you'll need to log into Meta's latest app using your Instagram account - yes, that means you'll need to make an account for Instagram if you don't already have one, although thankfully you can do this from the Threads login page without needing to open Instagram separately. Unlike most other social media apps, you can't make an account for Threads.