OK, now we have the ball rolling
What do you have currently?
Lets start with a camera/lens. I would recommend that you use a Digital SLR camera. You don't necessarily need a pro level model, the entry and mid level models are quite capable but you will want a good quality lens. You can get a good DSLR for less than $1000 (maybe closer to $600) for an entry level model. For a lens, there are many options...for a zoom lens, I suggest something with a maximum aperture of F2.8...they start around $500 and go up from there. You can go with prime (non-zoom) lenses and probably get more image quality for your dollar...but being able to zoom may be very handy, especially when shooting small kids.
You can do many things with just a camera and lens...but for setting up a studio and even for shooting outdoors...you may need lighting. Some people prefer to use only natural light...but this can be limiting and also can be hard when shooting kids who won't sit still. In a studio setting, you will really want to have your light off of the camera. You can choose between continuous 'hot' lights and strobe 'flash' lights. I really suggest strobe or flash lighting.
Again, there are different ways you can go. You could use hot-shoe flashes...which can be used on the camera. With some accesories or multiple units, they can also be used off camera. They don't have a lot of power but they run on batteries which is good for portability.
Alternatively, you can go with studio style lights. They are more powerful than hot-shoe flashes but then need to be plugged into something for power.
The range of prices is very wide here. Hot-shoe flashes can be very cheap for simple models and up to $500 each for top of the line dedicated models. Studio lights are not cheap either. Check
www.alienbees.com for some good lights at good prices.
Again, there are different ways to go. One light may be enough, if you know how to work with it. Or you could get several lights and have more options. You also need the stands and modifiers (soft boxs, umbrellas etc.). I would think that a set of lights for a starting home studio might cost you $1600-$2000.
You might also need/want a flash meter ($200+) and a background system. A basic background stand is about $100 and then you can go with seamless paper rolls or fabric backgrounds...prices will vary.
I'm sure I'm not making a comprehensive list here...just going off the top of my head. As you can see...there are many ways to go about this. It can be very simple or complex.