Math PhD in US vs Master + PhD in Europe
I got my PhD in Germany (Heidelberg) and then did a postdoc in the US. I think that was the right choice: In the US, you will have to spend a couple of years taking classes, satisfying your breadth requirement, and you will likely have to teach alongside your research. There is much to be said about this system -- including that it teaches you to teach, and that you get a broader overview of what mathematics is -- but it all gets in your way of doing research. On the other hand, in the German system, you will most likely have relatively light teaching duties (if any) and have time to specialize. This will help you find a research area, and it will likely mean that you are better qualified than many American applicants once you apply for postdoc positions.

What is going to be important is that you form the personal connections on the other side of the pond to get a postdoc position there. Postdoc positions are not just awarded by accomplishments alone -- much as we would like that to be the case -- but also based on who you know, who your adviser knows, etc. This is so because when you apply for postdoc positions, your resume is still relatively thin: A couple of papers, a couple of conference presentations. In essence, every applicant on paper looks equally good, and so things like what university you come from, who your adviser was, and what the letters of recommendation say all have influence. The biggest influencer is if you know someone at the university you apply to. In other words, build these connections: go to conferences in the US if you can, contact people there to see whether they want to collaborate with you, see if your adviser knows people there and can support you for a 4-week stay. All of these things are going to matter.

Second, see that you can teach a course for a semester. Almost all US graduate students teach courses, and postdocs generally have to do the same. As a consequence, having taught (and having letters of recommendation that speak to your abilities as a teacher) is a criterion that is often considered when evaluating postdoc candidates. Try to make the selection committees job easier by providing them something in this area.

Finally, make yourself knowledgeable about the application deadlines in the US. These days, they are often at the end of October or November, for appointments that start in August or September of next year, at the beginning of the next academic year. (One can think that that is crazy, but that's how it is.) Most applicants will not yet have graduated at the time they apply, hoping to graduate at the end of the spring (May) or even summer semester (August). In other words, don't wait until you have your PhD to apply for a postdoc position -- you might have a gap of nearly a year if you do!