To begin, I would want to offer my sincere congrats. 🥳
You should go to this page Presidency of Migration Management (PMM) (goc.gov.tr) with your login information to check and see if a tracking number has been generated for your residence permit.
If your unable to access your account or have forgotten your login credentials please contact your authorized agent or call 157.
After you have either received or had your SMS message from PTT stating that your card is on its way, you can consider the process complete.
You can check the status of your package by entering your tracking number at gonderitakip.ptt.gov.tr.
They will make one delivery attempt at the address you have provided with your application. However, if you are out and about and the delivery guy misses you, there is no need for concern on your part.
Simply remove the label from your door and bring your passport with you when you go to the PTT branch that is closest to you.