Contact the Jobcenter if you want to enrol in a course. If you are already working, doing an apprenticeship (Ausbildung) or having a degree recognised, you can contact the BAMF: firstname.lastname@example.org. There, you get a certificate of eligibility (Berechtigungsschein) for the course. You can turn to your employer as well, who will receive additional information from the BAMF.
Usually, the courses are free of charge. If you earn more than €20,000 on your own or if you and your spouse together earn more than €40,000 per year, you might have to pay €2.07 per class hour.
Passage Hamburg informs about job-related German courses (berufsbezogene Deutschkurse) and also offers courses themselves. email@example.com, +49 40 24874815 (Monday, Tuesday, Thursday and Friday, 8 am – 1 pm), http://www.wege-in-den-beruf.de/deufoev.html.
The Netzwerk FLUCHTort helps refugees with an uncertain residence status (for example a permission to remain while the asylum procedure is pending (Aufenthaltsgestattung) or a temporary suspension of deportation (Duldung)) to prepare for an apprenticeship (Ausbildung) or a job. There are various training and coaching offers. You can find more information here: https://www.fluchtort-hamburg.de/fluchtort-hamburg-50/traeger-und-teilprojekte.