Hello and thank you for your question!
As far as I understand, you were granted asylum according to section 16a of the Basic Law, right?
Handbook Germany describe your rights - including the issuance of a residence permit according to section 25 paragraph 1 of the German Residence Act - very well. Please follow the link for the details. My colleague Éanna explained in this thread the usual procedure after the final BAMF decision on an asylum application. You may send an application for the residence permit to your local immigration office (Ausländerbehörde) already and in case you can not cover your costs of living with i.e. an income, you can apply for unemployment money at the jobcenter as well.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike