What is the minimum equipment for setting up a photography studio at home?
First, you need some lights. They can be either speed lights, studio strobes or some kinds of continuous light.
I use studio strobes myself, and I can recommend those. One big advantage is that there are many accessories, light modifiers, available. I suggest that you start buying the lights, and from then on the other things, as you first will have to decide what mounting system you want, depending on the system your new strobes are for. Mine is Bowens, and that is supposed to be a widely used system.
So my advice is that you buy the lights first. But how many lights should you start up with?
You can actually go far with only a single unit, but having more will make it possible to do more advanced setups. I will suggest to you that you buy two strobe heads for a start. Then add one or two more later on.
But what accessories you will need, all comes down to what type of photography you will make: portraits, macro, still life, products, or maybe beautiful pictures of nature.
Studio light can be useful in all situations, except the latter, and the studio light will be beneficial in the other branches of photography mentioned here. The question is about what kind of light modifiers you shall use for the different types of photography. But then again; many modifiers will be useful in several different setups.
From a starting point, I assume that you have a camera and at least one lens.
In many situations you will need a background of some kind for shooting portraits. You can, if you have a wall, paint it. Black and white are maybe the most useful. Or you can use things like a cloth, a huge one like a bed sheet. Or you can go buy a real background. There are several possibilities like paper, vinyl, fabric, or different printed backgrounds. You will need some stands for both the strobes and perhaps the background. Then for a light modifier I will suggest you a soft box as a minimum for one strobe, and an umbrella for another. Notice that you can use the standard reflector on your strobe head also. And the umbrella is cheap though, and worth the price. Then you can add more things later.
For an upgrade I will recommend that you buy that third strobe light. Remember a stand and some kind of light modifier, like a strip soft box, or a beauty dish. Also, it is time to get a proper background, if you skipped that before. The things you want to buy, and what you need depends on what kind of photography you want to take. If you are more on portraits then you really should look at what you can find for making such. Like a system for backgrounds, and the backgrounds themselves.
If you are doing product photography, then, you may want to buy a special table for that. You can find them on Amazon, for reasonable money.
If you are more into still life, then a good tripod may be on your shopping list and some macro equipment, see the section about macro photography in this book. I look at several ways to do macro, with different tools. About lighting for still life, - add a stand for your light.
A second upgrade can be about things like a flash meter.
This is simply a useful piece of equipment.
You can buy more light modifiers, you most likely by now have an idea about what you may need.
Remember that things like software, memory cards, and printers, are very useful also. Try always to buy good quality items.
If you plan to do some photography at home, or in a kind of studio, then you can help yourself by having a power supply for your camera. This way you don't have to buy more (expensive) batteries and chargers yet still always run out of power. A power supply is a very useful thing. If you still want to use batteries, then you can buy extra batteries, and/or you can get a grip for your camera, where you can have two batteries installed, for sustained action.
But ... you still have to charge them.
So you now have your photography equipment.