How does the idea of race, ethnicity, and nationality impact one another?

A problem in dealing with the concepts in your question is that their definitions are subject to debate, have changed over time, and are different for different people.
"Race" is a difficult word, partly because the concept has been used as the basis for oppression, the enslavement of people, and genocide. Even in writings from as late as the middle of the twentieth century, "race" was sometimes considered synonymous with what we generally now call ethnicity, as in references to the "Latin races" of Europe. This would have meant people from the countries in which Romance languages are spoken: Italy, France, Spain, the French speakers of Belgium and Switzerland, and the Romanians. But what makes people a "race"? In our time, "race" is more scientifically defined based on genetic testing and on one's DNA. During the same relatively recent time, however, when race and language could still be seen as connected, one often heard that the world was divided into three races, "white," "black," and "yellow." This, of course, was incredibly simplistic. One would think that a commonsense view of "color" would be that it is a kind of continuum, a spectrum in which there are no cutoff points or clear divisions. In that case, the concept of "race" as a separation of humanity into distinct groups would seem to have little foundation.
The term ethnicity comes from the Greek word meaning "nation." Americans used to (and sometimes still do) use "nationality" to mean ancestry, which is the same thing as "ethnic background." This, ultimately, is based on the language your ancestors spoke. But how is this connected with one's heredity or genetic makeup? In Europe, throughout history, huge population movements have occurred again and again, and the language spoken in any given country cannot possibly have a one-to-one relationship with the genetic ancestry of the people who speak it. Italy, for example, was repeatedly invaded by Germanic tribes in late antiquity and the early Middle Ages. These people mixed with the indigenous population of the Italian peninsula, and modern Italians are descended from them. However, they speak Italian, a Romance language, though much of their ancestry is shared with Germanic peoples who speak German, Dutch, and the Scandinavian languages. The population of modern England is descended from a mixture of people who spoke Celtic, Germanic, and Romance languages because England was repeatedly invaded by the Romans, Saxons, Danes and other Germanic groups. The Normans were themselves descended from the Scandinavians, who had invaded Normandy and then adopted its French language.
In summary, race, ethnicity, and nationality are nebulous concepts with multiple definitions. Unfortunately, they have been used in nonscientific and mythic ways to divide rather than unite people. The Nazis used the false idea of a "superior" and "pure" German race or nation as the basis for the genocide of those they considered inferior. The different "race" of African people was used by Europeans to justify enslaving them. Countless other examples can be found through history of atrocities being committed based on concepts that have meant different things to different people.