Here's a guide to how I approached building a home gym for myself!
Talking about my case, I have been into weight-lifting and a bit of calisthenics.The essentials:
Squat Rack: There are multiple type of squat racks available, power racks, single stands. My suggestion: If you are into compound movements like bench press, squats, I'd suggest a power cage like the Decathlon Rack 900(which I have), or if you are low on space, the Rack 500 is also a good starting option. But you will need to upgrade pretty soon.
Bench: You need a bench no matter what, I use the Decathlon Bench 900. However, there are a lot of options nowadays.
Flooring: Gym equipment will damage your flooring if you are not careful. Get horse-stall mats or dedicated gym mats. I use these, ordered 6 of them. 4 for my squat rack, 2 for multipurpose + deadlift platform.
Barbell: Get a 7ft 20kg olympic barbell if you have been lifitng for a while. For beginners, I'd suggest a 5ft 10kg one.
Dumbells: Get adjustable dumbbells, that's the best bet. Or just plain 2kg dumbell bars, you can just add weight plates on it.
Weight plates: If you like steel and the clunking sound, get cast iron or steel plates like these. Or if you wanna keep things quiet and protect your floor, get the rubber bumper plates like these.
Machines: This is a tough thing to fit, what most people is to attach a pulley in your squat rack and perform cable exercises. The rack 900 has a cable machine in built with a pullup bar, so it serves really well.
Gym rings(optional): If you are into calisthenics, attach it in your squat rack, and you are sorted for dips, pullups, etc.
Tips:
Don't skimp on equipment quality, if you are buying good quality weight plates, clippers,etc. It will last you for years.
Start slow, don't get everything at once. Get the bench and weights, get the flooring, then go to the squat rack. Or maybe you can if you want though.
Prepare to take care of the gym every week. If you want to keep the space clean, you will need to vacuum, brush and wipe the flooring every other week. It's like your own gym.
Dedicate a separate space for the gym, I think that's pretty obvious xD, but it becomes a headache once you change houses frequently.
The pros:
The flexibility of an home gym is unmatched, you can workout whenever you want. No need to get ready, go anywhere.
Your family and friends can workout, it's a nice way to socialize at your place.
No waiting for machines and people.
You won't be able to go back to a public gym after having a home gym, because it's THAT good of an experience.
It will take you 2-3 years to break even the cost of the home gym but it's worth it.