There are different lights that can be used for studio photography. I will touch studio strobes, speed lights, continuous lighting, bulbs and LEDs.
Speed lights can, with the help of an adapter, make use of all the studio equipment, like soft boxes, beauty dishes and other light modifiers. Where speed lights, since they can run on standard batteries, are good for field trips or places where you have no power socket. If you need to bring a studio light setup with you, then speed lights can be what you are after. Or if you already have studio lights, then you can easily use your light modifiers with the speed lights. Just go shopping for the adapters. Then your speed lights can simply be your portable studio setup.
If you prefer real studio strobes, I have seen strobes from 180 Watt to 1200 Watt. As I personally have a very small studio, I don't need as much power to light up the scene. I have three heads of 300 Watt. And they are almost all the time on lowest, or (less than) half power.
Most light modifiers are made for studio strobes. And they are used by professional photographers. The strobes I have are at the cheaper end of the scale, but I think they are really great and they do the job for me, being a happy amateur having photography as my hobby.
They feel like good quality, and I am sure I will not soon exchange them. Actually, I am considering a fourth strobe head of the same make and model to the studio setup I have. The selection of light modifiers is huge, and many companies are competing, making the market really big.
Continuous light for video.
With the innovation of lights, especially LEDs, you can now find good and cheap continuous lighting that is really great if you want to use the light for both video and still photography.
You can also find lamps for use with traditional bulb-like lights which are newer and less power consuming, yet giving a nice light for your photography use. What you have to look at with continuous lighting is the temperature of the light.
Normally, I will not buy anything as a kit. You always seem to have to buy something in the kit that you really do not want. But, because I bought from overseas, I had to think about the cost of shipping also, and buying one item will definitely be cheaper on shipping than buying most things from different sellers.
What I found at Amazon.co.uk was "BPS 900W Flash Strobe Lighting". It includes 3 x 300W strobes, 3 stands, 2 soft boxes, 1 trigger set, 2 umbrellas, a free carry bag and some other accessories. The price as at the time of writing is £333.55. Then, you have to add the shipping cost.
These strobes are made by Godox but are sold under different brand names.
Browsing for studio strobes, I found that kit on Amazon suited me in a way. All the items in the kit were things I wanted to buy anyway. I wanted other things,but at lesser priority, like a barn door, and a snoot, but all the light modifiers that came with the kit were items I wanted, so it was an okay deal for me. I then later had to find the barn door and the snoot elsewhere.
Knowing only speed lights, and bulbs, I spent a long time browsing for the right strobes for my need - and I am fully satisfied with the kit I finally bought via Amazon.
I use the wireless trigger system that also came with the kit. There is a trigger that has to be attached to the hot shoe on the camera. With that follows a receiver that is to be connected to the sync port on the strobe. Then, if using more than that one strobe I will set up the others as slaves.
This works really well.
Above, and below; the trigger system, by Neewer
If I want to use the speed light, I will have to trigger the whole setup, using the small built in flash on the camera. This is needed for my speed light. It will not trigger being a slave in the ordinary way like what triggers the other strobes as slaves. But the strobes can be set to work with the TTL that my built in flash uses.
This will trigger all the studio strobes and the speed light. But it is not always so that I want the camera flash to shoot. So I don't use my speed light for studio light.
I don't need that extra light now, but if I will at some point, I think I will buy an extra studio strobe.
They are really good.