The first major law to take away Jewish rights was the Law for the Restoration of the Professional Civil Service. This law prevented Jews from being involved with public service and was put in place in 1933.
Not coincidentally, 1933 is also the year that Hitler became Chancellor and it was only a few months into 1933 and virtually all organized political resistance was suppressed. Thus, the Nazis essentially began moving against the Jews once there was no real political will to stop them (although there was certainly cases of Nazi resistance in general over the next 10 years).
The interesting thing though is that the Nazis had been discussing these actions well before they had total power. In their 25 point plan published in the early 20s, and disseminated by Hitler as their chief of propaganda at the time, the Nazis spelled out their intention to take away Jewish rights and separate Jews from Aryan society. During this time, the Social Democratic Party of Germany and the Communist party of Germany opposed the Nazis. The communist party was by far the largest political resistance to the Nazis. At this time in history, communism was growing all over the world and was the primary opposition to fascist government. The communists certainly believed in equality and therefore attracted those sympathetic to Jews. The Baum group was a Jewish resistant group that was communist.
Some basic sources https://en.m.wikipedia.org/wiki/German_resistance_to_Nazism
https://www.ushmm.org/wlc/mobile/en/article.php?ModuleId=10007901
More specific resource McDonough, F. (2001). Opposition and resistance in Nazi Germany. Cambridge, UK: Cambridge University Press.
As to the "registering" part:
There is considerable debate whether the Gestapo or the SD or the Reichssippenamt used the informations of the 1933 and mainly 1939 census to create a Judenkartei - "register of jews". This thesis is implied in Die restlose Erfassung by Götz Aly and Karl Roth. We should note that the book was planned as an argument against the 1984 census of the FRG.
There are some who argue against this, as the German authorities already had most of the data from tax files, the population registers, (every birth, death, marriage was/is recorded in Germany), partly even military files.
All in all, it's safe to say that by the time the deportations (from Germany to the East) began - 1941 - the Nazis had files of every Jew in Germany.
Many of the answers given help point to the silencing of opposition that occurred in Germany early in Hitler's time in office. I point to my post here to get an idea of how this was done and why opposition to it didn't occur. It was a process called gleischaltung that made Germany what seemed to be a unified national community. This unity can be overstated, but at least from the outside, it was quite an amazing feature of early Nazi rule.