Again, just like the Empire, they "took over" the galaxy in that they had the largest fleet. They publicly arose five years before Episode 7, but were making movements for eight years prior to that, with most of the Senate of the New Republic dismissing the legitimacy of how much of a threat they were, with only a portion of them supporting the Resistance. By the time the First Order publicly arose, it was too late to effectively counter it, and Starkiller Base in Episode 7 crippled the New Republic by using Starkiller Base to destroy the planet of the Senate as well as the New Republic fleet.
Yes, a lot of these details are in expanded universe material, but films like this aren't designed to bog the viewer down with a lot of minor details. Like, you can find out how lightsabers are actually constructed, but it's not important enough for the focus of a film to spend time having Obi-Wan explain all that to Luke. Pacing is crucial to a film, sometimes you just have to go with what info the movie is providing you and fill in the blanks yourself.
You're assuming planets in Star Wars are as densely populated as our Earth. Many clearly have far smaller populations, and some are practically devoid of intelligent life. Also, only a portion of a planet's population would be fit to enlist, and the Empire would require time, resources, personnel, facilities, etc. to train them. Only four years pass in-universe for the original trilogy. And remember, a lot of that time, resources, and personnel was focused on building the second Death Star, so that even further limits them from doing what you're suggesting.