When we entered Germany (now already one and a half year ago) my now husband (South Korean nationality) could enter and stay without visa for three months. Before this three months expired, we had to visit the Auslaenderbehoerde (Foreigners Office) at Friedrich-Krause-Ufer.
South Koreans fall under the same visa category like a few other nationalities (Australians, Israelis, Japaneses, New Zealanders and North,- Middle- and South Americans) who are privileged to change their status (Aufenthaltstitel) while in Germany.
You have to book an appointment by the internet. This link applies to all nationalities.
The first time we were foolish enough to go to the Auslaenderamt without taking a prior appointment. However we've been lucky because we came before opening time and waited for quite a while in the queue. Just shortly before the doors opened, the queue wasn't any longer respected - much to our frustration. Once the doors opened, we felt like being out in the wild (pushing elbows, running up stairs) people ran to get the few tokens they release on each day. We rushed to also get one too. Even these tokens normally are just there to make an appointment (we were told later) and not to clear your issue. However, we've been lucky. We had all the necessary papers and my now husband got a temporary extension from a friendly civil servant.
Unfortunately we had to extend this temporary visa once more because we didn't manage to get our marriage organized before the three months expired (this is another story of crazy German bureaucracy ). Even though we had an appointment which we had booked by internet a few weeks in advance, we had to wait a long time and only after I'd asked them what happened, we got called into their room. I think they simply had forgotten us and were embarrassed to say so. This time they were really unfriendly and we had to listen to some silly blablabla but anyhow, we got another extension of three months. Because we had the correct documents together. Note that you are not allowed to work with this temporary visa.
A few weeks later we did get married. YIPPEE!
Again we had booked the appointment at the Auslaenderamt in advance by internet and he got his 3 years Aufenthaltstitel (residence permit) without problems. This time there was again a friendly lady who served us. It's just so much nicer to deal with friendly people.
Make sure, you book your appointment and bring all the necessary documents.
And have a lot of PATIENCE and SMILE !