Scholarships Funding Studies in Germany
One of the reasons that many people give as to why studying abroad is difficult for them is the costs associated with doing so. I was lucky enough during my time studying abroad during my undergraduate studies to receive a scholarship from my college, which helped to defray the costs. The Deutscher Akademischer Austauschdienst (DAAD) is the German government's organization which exists to provide scholarships for foreigners to study within Germany, and for Germans to study internationally. If you are American but aren't a dual citizen of a European country (and therefore are ineligible for the ERASMUS program for European students studying in other European countries), then applying for one of the programs through the DAAD is the primary option.
The DAAD has programs available for both undergraduates and graduate students, which provide money for short periods of time (typically 4-10 months). There are options for research and studying abroad, though it's worth noting that the research grants will likely be primarily useful to those in STEM degree paths. If you apply for one of the research grants, you'll need to have your position in a lab or research group confirmed prior to the application. If you are studying abroad and are doing so in the context of an organized program at your home institution, they should be able to help you with the application.
For graduate students specifically, there is a time limit of 15 months since you started living in Germany to apply, meaning if you are taking part in a graduate degree program full time in Germany (not doing an exchange or time abroad as part of a U.S. based program), you should apply sooner rather than later. After you have been in Germany for 15 months, you are expected to apply to other stipend funds that exist in Germany which are primarily targeted at Germans (but do not require German citizenship to apply). If a scholarship from the DAAD could be useful for you, you should apply sooner rather than later, as some of the other programs may be more difficult to apply to for those who do not have a B1/B2 or higher German level.