In Germany you need to register your residence (Anmeldung), within a week of finding a permanent accommodation, at the local office of your community, in Leipzig you will have to do this at the Bürgeramt, the location of which depends on where you live. You will need to fill a form, show your passport and that of all members of your family, and a copy of your rental agreement. The copy of the form “Anmeldebestätigung” you will get will help you for many other administrative formalities, so keep it safe. If you happen to move within the city you will also have to register the change, as well as if you leave for good.
Then you might also need to get a resident permit. According to your country of origin you will have to get an Aufenthaltserlaubnis or a visa at the Ausländerbehörde. Check the website for what papers you might need to provide (all in German) and to download forms needed for your registration.
Work permits are issued by the Ausländerbehörde, to obtain the permit a Meldebescheinigung or Anmeldebestätigung, an Auftenthaltserlaubnis and a certificate from an employer might be necessary. As every individual is in a specific situation, the best is to contact the Ausländerbehörde to know what your situation requires.