The best portrait focal lengths are either 85mm, or 105mm ,or 127mm, or 135mm, depending upon who you ask. Even pros disagree about this, and it is largely a matter of personal taste. If you got a kit lens with your camera it may include such focal lengths. Usually the kit lens is an 18-55mm, which in DX format is a 27-82.5mm. The 55mm setting gives 82.5mm field of view on a DX camera, and that is almost 85mm, so it can produce some passable portraits. If your kit lens was a 55-200 that gives the 82.5-300mm field of view, so portrait lengths are covered in that. Still, a zoom lens is softer-focus wise-than is a prime (single focal length) lens. For great portraits use a prime lens.
Nikon's AFS-DX Micro-Nikkor 85mm F/3.5G ED VR lens ($529.95 new) would provide a 127.5mm field of view on a DX camera, and that is a good portrait focal length. It is also a true (1.0X, full life size) Macro lens, and has 9 aperture blades...which produces good 'Bokeh' (background blur). If you can afford that 85mm lens, get one. You may save a bit by getting it used, or factory refurbished. Be sure to get the right one. Nikon makes several 85mm lenses which are FX (Full Frame) items for their professional cameras, and they are more expensive. The one I mentioned is their only DX format 85mm lens at this time. Until you can afford the 85mm lens, you can get by with any zoom which has 55mm, or above, in its range.
The 55-300mm will be as good as the 70-300, generally, but either may have some benefits, or detriments, the other doesn't. I haven't done side by side comparison tests, so I can't give details. Any of NIkon's flashes will be good. The more expensive ones have more convenience, and higher power, so they can be used from further away. I use an SB-300, with a cheap, foldable, soft box attached ( an Opteka I got on Amazon for about $7.00). I shoot from about 6 to 8 feet away, and it works fine. The head is tilt-able. It does not have all the whiz bang features of the costlier models, but I don't need them. The soft box diffuses, and softens, both the light, and the shadows, making things less stark, and glarey.
For general images of the kids, any zoom lens can do, and be convenient to let you change focal length at the twist of a wrist. 18-300mm would be handy. The 55-200mm can work. It is a matter of personal taste. For really memorable portraits of the head & shoulders, and head to waist, variety, get that 85mm lens, and a softbox for your flash. If you want to use several flash units-one in front, one on the side, one behind as a hair light, etc-get a costly flash which can be triggered by a main flash. If you want the SB-600, it may still be found in stores / warehouses, if you look hard enough. Check around, and see. If you can't find one, go with something else.