I shoot mostly family/documentary/kid style. Also moved from Pentax to Nikon D7000 then to D600.
Most importantly i set the AE-L button on the back to AF-ON. No AF on the shutter button!
Then i set the AF system to use AF-C.
This is my basic setup for AF operation. I focus by holding in the AE-L button on the back with my thumb. The camera will concentrate on focusing constantly while i can focus on composition. I use the shutter button only to take a picture or as many as i want. I can hold AF with my thumb and keep af point over subject, or i can press it once, then just shoot if distance to target doesnt change.
As far as AF point modes i mostly use single point. Sometimes i use 9-Point area, and sometimes i use 3-D track. Your D7000 will do a good job tracking skin tones, your kids clothes etc etc, in 3D-mode. I works great under good lighting. Use striped clothes on your kids and you camera will have no problem
Still, the most important setting would be AF-ON to the back of the camera (AE-L button)... It may be stragne and wierd at first, but when you get into it you'll most definitively will like it.
This is what i like the most about Nikon vs Pentax. The AF system.
Look at my profile picture... I shot it through a mirror, AF-On with thumb and AF set to AF-C and AUTO. I aimed as good as i could and the camera focused on my face @ f/1,8 and it nailed it perfectly. Sometimes AUTO mode works really good focusing on eyes/mouth/nooses, try it.