We've spent 5 years raising our Son here in Germany. The best way to introduce your child to the culture is by "throwing them into the water" i.e. having them participate in public german school. We moved to Germany during the beginning of US summer vacation. This enabled our Son to interact with German children, get used to some of the culture & habits, before entering the school system later on.
Before I go on I want to mention that if you're a legal resident in Germany (living there on a valid visa), then the legal guardian is automatically provided with KINDERGELD (child raising budget) for each child. It's only around 200 USD per month, but it definitely helps, that's for sure. Guardians receive such KINDERGELD until the child reaches age 18.
If your child is in 4th grade or below ... REQUEST FOR YOUR CHILD TO BE KEPT BEHIND ONE YEAR in the new school. I cannot stress the importance of this enough for anyone who plans on making Germany a long time or even permanent home. The German school system has three different career levels of education. After fourth grade you may enter fifth grade *HAUPTSCHULE* which most kids leave after 8th grade in order to pursue a three year career training education for basic professions (Construction, Mechanics, etc.) Alternatively, after fourth grade your child may be placed in *REALSCHULE* (standard for most parents) which will provide a better career path at the end of Realschule, higher paid 2 - 3 year apprenticeship options, etc. Finaly, the third alternative for those who are certain that their child will pursue a university education (not even remotely as important or costly as in the US), a child may get into *GYMNASIUM* which is completed after 12 years, just like the US High School System.
There are several reasons why it is critically important for a foreign child to repeat a grade before entering Realschule or Gymnasium in Germany. In Germany it's considered somewhat of a privilege to be part of the better educated who went to either REALSCHULE or GYMNASIUM. Consequently there's an expectation of participation & performance. It's not too much of an expectation, unless of course you don't speak/comprehend the language well enough. Then the perfomance expectation becomes a very real burden on the child. So by having your child repeat a year ... which can easily be taken care of with a written request to the principal ... hopefully he or she will be learning mostly things that he or she is already familiar with anyway, thereby making the job of learning during the first year much easier, while also being able to focus much better on learning the German language itself. Language skills are vitally important because if the child's performance is too low, the school may legally recommend for that child to attend HAUPTSCHULE instead, which could greatly diminishes the amount of career opportunities later on.
So repeating a year voluntarily ... is actually a good thing! 
Last but not least ... I would never recommend sending your child to Private or International school here in Germany. Not just because of the cost, but also because it makes integration into the German culture so much more difficult & time consuming for your child. We moved with our Son at the end of his 4th year in US school. He then visited in a class at his local school for a few weeks before their summer vacation actually started ... just to get a feel for school and to meet some of the local children in a more controlled environment. He then repeated his 4th year in German school, at our request, to provide him with the best *FAIR* opportunities when that important switch from 4th to 5th grade takes place. As a parent, you have very little say in the German school system. If the belief is that your child can't handle REALSCHULE or GYMNASIUM, then he or she will be recommended to attend the Hauptschule. So do your child a favor, by allowing him or her to repeat a year while learning the language at the same time.
Our Son loved it. Children are much more versatile than us adults ...