After you have been assigned to a federal state ("Bundesland"), you will first be accommodated in a reception centre ("Erstaufnahmeeinrichtung"). You will usually stay there for a minimum of six weeks and a maximum of six months. This accommodation is located near a service centre of the Federal Office for Migration and Refugees ("Bundesamt für Migration und Flüchtlinge"), which means it is only a short way to file your asylum application ("Asylantrag"). After a period in the reception centre, you will move to a different accommodation. Depending on the Bundesland or municipality ("Kommune"), this could be a communal accommodation centre ("Gemeinschaftsunterkunft") or a flat ("Wohnung"). You can also stay with relatives if they live in the same Bundesland, as the case may be.
Whether or not you are allowed to move out of a communal accommodation centre depends on the Bundesland in which you live and your personal situation. If you are allowed to look for a flat for yourself, you first have to ask the social services department ("Sozialamt") how high the rent is allowed to be.