Private gym is to open up a space for fitness in the owner's own home. What equipment to equip depends on your own needs, your usual fitness habits and your final weight. Don't blindly pursue more and perfection. Many devices have the same effect. As long as the methods and actions are proper, too many devices are not needed. I'm an assistant coach. I suggest you find a friend in this field and ask him to help you choose equipment and teach you how to use movements according to your body shape and specific situation. There are many brands of equipment, the more expensive the better, and the protective measures are good for home use. Generally, a bench press, a short barbell and dumbbells with different weights. A bird chest expander and a leg bench press. If you are fat, you need a sit-up device.

Basically, the muscles of the body can be trained. I hope this helps.