If your domain is not in the list of supported top level domain you can't transfer it. The reason being AWS has to communicate with TLD do that transfer. If your domain is supported, the detail steps to transfer the domain is given at
List of support domains are ...
I've got a domain on a registrar, which is in its grace period. I'd
like to move it to a new registrar, so I think releasing it will be
the best option. I had the impression releasing a domain name makes it
available for registration again, but I have a worrying warning from
my old registrar when about to release it
The terminology you are using is ...
A registrar detags a domain because for some reason they are assuming you don't want to renew. The detagging ensures that they are not billed by the registry for a domain you seemingly don't want.
There don't seem to be any information on whether the domain becomes available immediately for registration.