Cashiers aren't automatically trained as GSAs as part of their Target career. You need to be selected for the position. You might cover the lanes a few times as a cashier, but the majority of your training will take place after you're signed off as a GSA. You'll get access to the GSA/GSTL key once your training is complete.
If you want to develop for the position, talk to your GSTL and ETL-GE and let them know your interest. Then ask them for an opportunity that you can work on. Your leader will love this, as TMs taking initiative, wanting to develop and wanting to work on their opportunities is awesome. Give it a few weeks and make some progress on that opportunity, then talk to them again. That should get the ball rolling.
If you want some anecdotes on how long it took various TMs to be offered the GSA position, check this thread out:
How long did it take you to become a GSA?