There was a variety of opposition to Hitler and the Nazi Party during their rise. I assume you know of the Nazi's Sturmabteilung, or SA, a paramilitary organization that worked as the militant wing of the party. The Nazis were not alone in having such forceful wings. Many other paramilitary groups existed in the Weimar years, which openly opposed and, at times, fought directly with the SA in street battles. One such group was the Communist Party of Germany (KPD) aligned Roter Frontkämpferbund (RFB) or Red Front Fighters Alliance. Parties would also provoke confrontations by having rallies or marches in neighborhoods known to be under control of the Communists or Nazis or what have you, with the intent that the opposing side wouldn't let such rallies go unopposed. Thus, these would often result in brawls.

Like the Freikorps and other right-wing paramilitary groups in Weirmar Germany, the RFB and other left-wing groups had many veterans of WWI. So some members were experienced soldiers. Their purpose included defending party events or meetings, defending and policing communist-controlled neighborhoods, and also open confrontations with the rival parties' militants (including the SA).

There are, of course, the voting records. The Nazis never achieved an outright majority in any of the votes prior to their coming to power. The KPD, SPD, Centre Party, and others received significant support. While not the direct opposition you may be looking for here, it is worth including the electoral opposition to the Nazis. Of course, voting is not always the most effective means of stopping political opponents, hence the militant groups.

As for what happened to these opponents of Hitler and party, most were either imprisoned, killed, or fled. The RFB and other groups were banned, either in the final years of the Weimar Republic, as well as by the Nazis. As such, some were executed or imprisoned. In fact, the first concentration camp, Dachau, was built specifically to imprison political opponents (especially communists). While we remember the camps because of the crimes against humanity committed against the Jews, Roma, prisoners of war, homosexuals, and disabled, other victims included political opponents. Konrad Adenauer, who later became West Germany's first chancellor and a founder of the Christian Democratic Union party, opposed the Nazis and was imprisoned several times during the Nazi reign. So not everyone who opposed the Nazis was killed.

As for fleeing, many left-wing and centrist Germans fled Germany in the 30s to avoid fates such as these I have described. Many German communists who fled would later return to what became East Germany. Walter Ulbricht, who later became General Secretary of East Germany, was one such communist who lived in exile in France, Czechoslovakia, and the Soviet Union.

I think that's where I'll end this. If I haven't answered your question to your satisfaction, or if you have additional questions, please feel free to say so and I can try to answer!