I mainly assist people who have a residency status in Germany already so the following is based on my more limited knowledge of the accommodation situation during the asylum process.
A person is obliged to live in a refugee reception centre (Erstaufnahmeeinrichtung) for up to 18 months. If this person has a child, then the maximum length of time they have to live there is 6 months. After the period of time living in the first centre, the person is assigned to live in a particular refugee accommodation (this is called Zuweisung in German and you have to be notified in writing about which centre you are assigned to).
You can ask to be assigned to a particular accommodation either before you get this letter or once you receive it. You need to have concrete reasons for this. A common reason people are allowed to move to a specific centre is if they have family members there. Having the opportunity to work, study or do an Ausbildung is also a good reason to allow this move.
You have to make an application in writing explaining where you wish to move to and why. You will also have to provide evidence. It is not enough to write that you will have better chances of finding work or know people who can help you find a job. You’ll need to show you have permission (Arbeitserlaubnis) from the immigration authorities (Ausländerbehörde) and have a job offer.
Here is some detailed information I translated with an online tool about this procedure. This information is about a particular federal state of German (Niedersachsen) so the office responsible will differ depending on where you are:
In the centre you are currently staying, there will be social workers who should have experience in writing such applications. Let us know if you have any other questions.