Lighting is complicated! The first thing you need to decide is if this is for stills or video. In theory, you could do both with the same lighting, but for stills you are still better off going with flash/strobe lighting instead of constant lighting. We will start with stills. 
Now is a very fortunate time for photographers as the cost of lighting has come down tremendously in the last several years. You will find that the Chinese brand Godox is very popular for budget conscious shooters. That is a good choice to start with. But, if you are in the US, there is a better option. Adorama has a house brand called Flashpoint that is the same Godox products rebranded. With the advantage that you are now dealing with Adorama for warranty and support. So we will be looking at their offerings. 
For home/studio use, the starting point would be the Flashpoint BLAZ300. This is a wall powered 300 watt/second light that has the receiver for the Flashpoint wireless system built in. 300 watt/seconds refers to the maximum output of the light, and that is a good place to be. You can get 600, 800, 1000, and 1200 w/s lights, but I find that 300 is usually enough in most conditions. These retail for $109. 
The only real downside to the BLAZ300 is that they have a six stop range for power. To give an idea:
1/1 - 300 w/s, 1/2 - 150 w/s, 1/4 - 75 w/s, 1/8 - 37.5 w/s, 1/16 - 19 w/s, 1/32 - 10 w/s
You will find other lights that will drop to 1/64, 1/128, or even 1/256. In most cases, this is not a huge deal breaker. Where it will be an issue is when you want to shoot really wide open apertures in a semi bright environment. You could run into a situation where you can't turn down the BLAZ300 enough and the image will be overexposed unless you close down your aperture. But given the price of these, that is a reasonable tradeoff.
These lights will be controlled remotely from the top of the camera using one of the Flashpoint triggers. They are solid and reliable. Stands and light modifiers are next. There is no shortage of options. Starting out, I would recommend something like the kit above. Three strobes, three light stands, two 24x36 soft boxes, 1 umbrella, 1 reflector, a trigger and a rolling case. The three light kit is $550. It also comes in a two light kit for $420. Either will be enough to get you started with great results. 
The other interesting option is speedlights. You can think of these as the traditional flash you have seen on the top of the cameras. But these have the Flashpoint wireless receivers built in, so you can place them anywhere. This is the Zoom R2 and you can get these for $65. These have the opposite problem of the BLAZ line. They can drop to 1/128 power, so you can always get the power low enough. But it will be possible to run out of light. If you are trying to shoot an F11 portrait in a dark room at ISO 100, full power may not be enough. You would have to raise your ISO to get the correct exposure. For reference, these are somewhere around 65 watt/seconds. 
On the other hand, what these lack in power, they make up for with versatility. You can definitely use them for indoor studio setups, but they are also great if you want to do something outdoors. They use the same remote triggers as the rest of the Flashpoint lights. Two or three of these with stands and soft boxes would also be a great start.  Plan for $200 - $300 for stands, modifiers, and trigger to get started. 
Video Lighting
For video, you will want to go with LEDs. There is no end of options for this, but I like the Flashpoint FV150 pictured above as a great starting point. That is, with one caveat. Currently it is on sale for $200. It's a very good deal at that price. At it's regular price of $350, it's not as attractive. A couple of these is a great start to a video lighting rig. 
At a moderately lower price, you have something like the Godox SL60W, which can usually be had for around $130. This is not as powerful as the Flashpoint light above, so you may have to raise ISO, but your camera can handle that without issue, so it is still workable. These two would also work well together with one FV150 as a key and two SL60s as fill or rim lights. 
It is also worth mentioning modifier mounts here. Both of these lights use the Bowens S mount to attach modifiers (reflectors, soft boxes, etc.). That is the same mount used in the Blaz still lights and the mounts you can get for the speedlights above. This is something to consider because if you can match the mount on your still and video lights, that means your modifiers can do double duty. 
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