The first wave of emigration happened after the Reichstagsbrand/ burning of the Reichstag on the 27 February 1933. The Reichstagsbrand was used as a legitimisation for the Nazis to implement an emergency legislation, the "Notverordnung zum Schutz von Volk und Staat" which enabled them to imprison political opponents. Further the "Gesetz zur Wiederherstellung des Berufbeamtentums“ (Law for the Restoration of the Professional Civil Service) got implemented on the 7 April 1933. This law officially banned Jews, as well as political opponents, from all civil service jobs. There where exceptions for Jews who fought for Germany in WWI, but even this privilege ended in 1935. There was also a law introduced in 1933, the „Gesetz gegen die Überfüllung deutscher Schulen und Hochschulen“, which made Jews unable to visit higher schools and universities.
So in 1933 it was more their livelihood than their life that was threatened. At this point, it were mainly highly educated Jews and Intellectuals who left Germany.

The second wave of immigration started in autumn of 1935. The "Nürnberger Gesetze“ (Nuremberg Laws) were introduced on 15 September 1935. One law was the „Gesetz zum Schutz des deutschen Blutes und der deutschen Ehre“ (Law of the Protection of the German Blood and German Honour), which forbade marriages and sex between Jews and Germans, as well as the employment of a german woman under the age of 45 in a Jewish household. The second law, the Reichsbürgergesetz (Reich Citizen Law), basically declared that only Germans and those of related blood were able to classify as citizens. Everyone else automatically lost their citizenship and was classed as state subjects. This was the point, where the Ariernachweis/Aryan certificate was officially implemented. While this seems like a really big red flag in hindsight, there were quiet some Jews, who hoped the Nuremberg Laws would help them in the long run. Some orthodox circles even supported the part that forbade a marriage between a Jew and a non-Jew. Others believed that their life in the future maybe wouldn't be easy, but Germany as a constitutional state would still be a place worth living. Many felt a strong connection to Germany and their home towns and were unwilling to even thjink about fleeing.

Leaving Germany as a Jew wasn't easy, even thought one of the stated goals of the Nazis was that all Jews should leave Germany. Jews were required to pay up to 90% of their wealth as a tax(the Reichsfluchtsteuer) for simply leaving Germany. Transferring money overseas was a punishable crime. This made it significantly more difficult to start a new life somewhere far away from home. An exception was Palestine, due to the Haavara Agreement.

After the Novemberprogrom (Crystal Night) on the 9-10 November 1938 many more Jews wanted to flea Germany, as it marked the trasition from discrimination to systematic persecution. This was he point, where most if not all Jews understood, that there was no future for them in Germany, no matter how much they would try to assimilate. But at this point, there were almost no countries willing to take them anymore. Nonetheless, more than 40,000 Jews were able to leave the country in 1938, in 1939 it were around 78,000.

With the start of the war emigration was almost impossible. Only 30,000-35,000 Jews were able to escape.

In Oktober 1941 the Auswanderungsverbot (ban on emigration) got implemented. After that Jews were not able to leave the country on legal ways and even illegal ways were nearly impossible due to the war related events in and around Germany.

Sources:

Mommsen, Hans. Der Reichstagsbrand und seine politischen Folgen, In: Vierteljahrshefte für Zeitgeschichte, 12 (1964), Heft 4, S. 351–413.

Aly, Götz (2014). Why the Germans? Why the Jews? Envy, Race Hatred, and the Prehistory of the Holocaust. New York: Metropolitan.

Jünger, David (2016). Jahre der Ungewissheit: Emigrationspläne deutscher Juden 1933–1938, P. 238-253. Göttingen