The overwhelming majority of foreign visitors will never deal with issues of open racial discrimination or racism in Germany. Large cities in Germany are very cosmopolitan and multiethnic with large communities with origins from all continents and religions. Germans are also very aware and shamed of the historical burden of the Nazi era and are usually open-minded and tolerant in contacts with foreigners. Non-white visitors may get an occasional wary look (particularly in Eastern Germany), but not to greater extend than in other countries with predominantly white population.
Public displays of overt antisemitism are strictly forbidden by laws that are very much enforced. The Hitler salute and the swastika are banned, as well as the public denial of the Holocaust. Violations of these laws against racism are not taken lightly by the authorities. There is no such tolerance for things like "I was only joking about the matter", and it is considered very rude and tasteless behavior by most Germans.
The situation may be different in some predominantly rural parts of Eastern Germany (including the outskirts of East Berlin). The feeling of being left alone with widespread underemployment and unemployment and the desperation caused thereby can lead some people to xenophobia ("they are stealing our jobs") and therefore racism, making them easily influenced by right-wing groups. As a result there are more incidences of racist behavior than in the West with a few incidents of violence. Most of these happen at night when groups of drunken "Neo-Nazis" look for trouble (and solitary victims) downtown or near public transport stations. The anger of these groups is directed against anything which is different. Hence, it might not only affect non-white visitors, but also homeless persons, West Germans and people with alternative looks such as Punks, Goths, etc.
Large cities like Berlin, Hamburg, Munich and Cologne have a significant Turkish, Russian and Arab population. Especially on weekend nights, some of the (mostly poor and unemployed) youths gather at underground and railway stations in certain districts and look for trouble. Use your common sense to avoid them at night and do not travel alone (especially women) as these youths can easily be provoked and sometimes will attack you without any reason at all. By the way these groups of antisocial migrants are a major reason why many Germans have an aversion and fear of South European and Asian foreigners living in Germany, although most members of these communities are peaceful and integrated, just like most of the Germans delimit from Nazi-Germany. Certainly the skeptical attitude against Arabic-looking peoples ordinarily disappears quickly if they show good behaviour. Visitors from other regions are usually unconcerned.