When Hitler took power in 1933, he didn't actually start with the Jews, & they weren't kidnapped off the streets. The first concentration camp was at Dachau, & the first prisoners were political enemies - trade unionists (union organizers and members), Socialists & Communists. [Conveniently, not only were these groups Hitler's political enemies, each was a group used to working with it's own large membership to accomplish their goals; by imprisoning these people, Hitler not only neutered the political opposition, but he struck a heavy blow against civilian resistance as well.] Originally, they were simply arrested & disappeared, but yes, eventually the Gestapo became increasingly aggressive & did plain kidnappings.

Dachau was originally intended as a prison camp, but it needed to be expanded very quickly to accommodate all the newly arriving prisoners, & the ever-efficient Nazis (remember, they came to power because of the onerous penalties inflicted by the Treaty of Versailles, - the world was in the Great Depression, so the government really couldn't afford to spend any money) decided the best way to build out the camp was to not-spend much money doing that. Instead of hiring outside workers (who might also bring out word of conditions in the camp, be inclined to transmit messages/contraband in & out of the camp, & potentially hear messages of solidarity from union workers), they decided to have the prisoners build out the camp instead. And why pay the prisoners, when they should be glad to be getting their lodging & food provided for them? The camps were destined to become slave labor camps.

Some companies, still in the throes of the Depression, & with some portion of their workforce now inside the camps & no immediate candidates to replace them, asked the German government if they could hire workers from the camps. The government agreed. The companies got cheap labor & stopped struggling in the Depression, the government got the money to spend on governing, & life improved for those outside the camps - you just had to ignore the camps & slaves themselves. And over time, other industrial-scale companies that didn't have skilled workers in the camps turned to hiring other workers from the camps; if they didn't they couldn't compete in the marketplace.

Meanwhile, having neutered the political opposition, the Nazis started turning to "undesirables". Even here, the immediate targets weren't the Jews; the next targets were gypsies (Roma & Sinti), gay men, & Jehovah's Witnesses, all of whom were thought to be "soiling" German civil society. Lesser targets who didn't always end up in the camps included immigrants & the homeless; & later targets also included people with mental /physical disabilities, & in some cases just abnormalities (ie, dwarves).

Please note that I'm not saying that the Jews weren't being targeted by the government & civilians, just that in the early days of the Hitler regime, those actions were a series of increasingly restrictive laws & increasingly-frequent/severe stochastic attacks encouraged by the government. However, Jews originally weren't interned in the camps to the scale they were imprisoned later on; they were "encouraged" to emigrate. The major turning point against the German Jews is generally accepted as Kristallnacht (Night of the Broken Glass), in 1938, five years after Hitler took power.

Remember that, in the 1930's, the majority of formal political & economic organizing was done by men, so the majority of the original prisoners were also men. The government did imprison women & children, but not originally at the scale later seen, so the imprisoned women & children were a necessary burden to "cleanse" German society. It was only after Germany started World War II by invading Poland (1939), that things turned dire.

In the newly-occupied territories, yes, the occupying forces kidnapped & killed/imprisoned those suspected of being enemies of the Germans, or just against what Germans thought their society approved of. They established camps for those they targeted in occupied territories, & forced occupied Jews into ghettoes, which were eventually "liquidated" either directly (mass killings) or indirectly (the people in the ghettoes sent to camps).

The camps became a problem. Established quickly, there were too many people to properly feed or care for. Medical attention was essentially non-existent. Conditions in the camps meant that diseases periodically ripped through the populations there. Since the Germans determined to imprison everyone in their target groups, that included young children, the very old & infirm, the chronically ill, etc. The camps became crowded, with a noticable percentage of people who simply couldn't work to "earn their keep". In merciless fascist logic, it was more "efficient" to kill those people and direct whatever meagre resources they were willing to allot to the camps to go their slave laborers.