Gary is right on several fronts. Portraits require imagative lighting to produce a wow picture. That includes the use of natural light, natural light with artificial fill in, and studio lighting. If you want to take pictures of the kids, a good camera, a good lens, a good flash will cover most of your needs. But if you are going to do this for money, a lot more is needed; including an education. I would suggest getting knowledgeable in the use of light. I'm sure that a local community education program can provide courses towards that end. In addition, there are a number of books that address this very need.
As far as a camera system is concerned, Gary is right again, that a number of brands would be fine. Much depends on the extent of the system and comfort level you have with that system. Nikon, Canon, Sony, Pentax all provide enough of a system to meet your needs. This includes camera, lens, and flash. Canon and Nikon are the most extensive systems, with a large assortment of lenses, cameras, and flash. All of the manufacturers have several levels of cameras that are designed to meet a consumers budget. This leads to my first question; what do you have in mind to spend?
Assuming you have the budget to do this system, an example system (and only an example system) would be a Nikon D 7000 with 18-105 kit lens ($1200) for general use, a 85 1.8 lens ($500) for head shot portraits, a SB 700 flash ($327) for additional lighting, and a Manfrotto 55xprob tripod with 498 RC2 ball head ($239.00). Total cost: $2267. As I said, this is only an example of a base system for portrait photography. If you just want to do your own kids and maybe some neighbors as a favor, you could do it for less. As you said the choices are many, narrowing those choices depends on your ultImate goal and your budget.
You should know that I am a Nikon owner. I know the Nikon system best, which is why I used a Nikon as an example. Canon has a similar system with the EOS 60D. Cheaper cameras would include the Nikon D5200, and Canon T4i. Again, what is your budget?
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