Continuous lights aren't great, because you would be dependent on your shutter speed for exposure. This would mean that unless you had A LOT of wattage (which would get hot), that you would likely be using rather high ISO and/or slower shutter speeds. And you don't want slower shutter speeds when shooting people.
That is why we like flash/strobe. The burst of flash is much faster than a typical shutter speed, and that freezes movement and gives you sharp photos.
So going in the flash/strobe direction, you have a few choices. The first is hot-shoe type flash units. They can be used on-camera, but can also be put on stands and used with umbrellas or softboxes. The pros are that they run on AA batteries and are small, thus you can use them anywhere. The cons are that they aren't as powerful or quick to recharge (and the good ones are quite expensive) and they don't really have a modeling light.
The next option would be 'studio strobes'. These are typically units that you plug into a power outlet (although, they can run on battery power). They are usually made to attach to all sorts of accessories like softboxes, umbrellas, snoots, barn doors etc. You can get ones that a lot of lighting power, although you probably won't need much for a small home studio.
They typically have built-in optical triggers but you may want one set of remote triggers (or use a cord) to fire them.
There are plenty of 'cheap' kits out there...which might be OK. But if you have plans for the future, the cheap kits probably won't last all that long...they either fall apart or you would quickly outgrow them.
This is about as entry level as I would recommend.
Flashpoint 3 Light Strobe Outfit FP3203