Covered in more depth by @Yevgeny Brikman but specifically answering the OP's questions:
What's the best practice for actually managing the terraform files and state?
Use git for TF files. But don't check State files in (i.e. tfstate). Instead use Terragrunt for sync / locking of state files to S3.
but do I commit tfstate as well?
No.
Should that reside somewhere like S3?
Yes
