Well, I got interrupted three times in composing an answer so some of this will be redundant with the intervening 6 posts or so. I used to work freelancing in a dozen or so studios doing product and model photography in studio and on location.
Hot lights (continuous tungsten sources) are great for seeing exactly how your lighting is working, but they are often underpowered for use with slower portrait films, and if you get hot lights that are bright enough to stop motion and use middle apertures, it often becomes painfully hot and bright for the subject. You can give models a 'sunburn' pretty easily with a quartz light and no scrim or diffusion. You might find that you quickly outgrow a beginner's SV three light starter kit unless you keep your project small and only shoot one person with lights pretty close in to the subject.
Studio flash and hot shoe flash often don't give a very good sense of the lighting balance with multiple heads in use, and their modeling lights sometimes don't track with your adjustments to the flash intensity. Hot shoe units give you no modeling lights at all. Studios that used flash with film went through polaroid materials by the case to check lighting.
Starting out with three light kits can become confusing and complex for someone new to lighting. Search the web for 'one light portraits' for a lot of ideas. Learning to use one light and fill with reflectors is both a great way to learn, and a good way to guard against unnatural looking 'overlighting'. Home made scrims (rip-stop white nylon), diffusers, foam core, aluminum foil, silvered car windshield reflectors, and many other methods are documented online. Reflectors can often substitute for a second, third, or multiple light sources if you use them properly. The Strobist blog and DIYphotography websites (and others) are good for ideas, and the hardware, fabric store, and craft store are good sources of materials. If you start with just one light, you can afford a more powerful unit on the same budget and fill with inexpensive reflectors.
As with cameras, lighting hardware is not a substitute for understanding and ingenuity.
I'd also very highly recommend any edition of the book Light, Science, and Magic by Hunter and Fuqua (and Biver in later editions). If you read it, you'll be well on your way to understanding working with light, and know more about what you'll need for your purposes. A used one from Amazon for about $18 would do perfectly well. The behavior and physics of light hasn't changed much in the last couple of years.
I wouldn't make any specific recommendations though, unless I could sit down and ask you a lot of questions. There are just too many variables involved to answer such a generic question. How old are your kids? Will they sit still to pose? Do you want to spend a good deal of money on multiple lights up front, or are you willing to get one good brighter light source and build a larger kit as you need it?
... and many more
Lee