To start an Ausbildung you have to find an employer (e.g. a company) that offers a so-called Ausbildungsplatz. There are websites where you can search for them. For example these ones:
You can also send your application directly to a company (that's a so-called "Initiativbewerbung").
An Ausbildung usually starts in August or September, but of course the deadlines to apply are earlier (that depends on the field you want to work in and the company).
After you found an employer, you will sign a training contract. You need the contract to register at "Berufsschule" (a school that you have to go to whilst you are doing the Ausbildung).
In addition, I will link our experts in educational topics: @BildungsberaterIn of LernLaden, Fadi of @Arrivo-EMSA and @KironHelpDesk of Kiron.
Also, @DavidJ at workeer and @Franziska_jobs4refugees at Jobs4Refugees can give you valuable advice on this topic and their programmes that connect employers and refugees.