Only a DSLR is up to the task what you are asking for. The big "but" is that it will require you to learn more about photography. When you use the appropriate lenses for low light photography, the depth of field is very shallow. Such lenses have the ability to give the blurred backgrounds which are quite fashionable, but makes understanding and using AF much more critical.
A strong zooming capability is not something that is easily or cheaply obtained with a DSLR. Because A DSLR's sensor is much bigger, much larger lenses are required. No single lens would cover the same zoom range as the HX100V, nor be able to also work in low light. For most people it's best to start with the "kit" lens, which is usually an 18-55mm. The second most popular lens adds telephoto, usually a 55-200mm or thereabouts. A typical choice for low light is a 50mm f/1.8 lens. There are lenses that have zooming capability and also work in low light, but such lenses are very expensive; you can easily spend thousands of dollars.
As for a specific recommendation and sticking with Sony, the A57 with kit lens, a 35mm f/1/8, and the 75-300mm telephoto lens. That would be about $1000. Canon's similar offering would be the T4i with kit lens, a 30mm f/2.0, and the 55-250mm lens.
I have 2 very active toddler twin boys that never sit still. I have been using a SONY Cybershot DSC HX-100V, but I want something better. I am looking for a camera that can take faster pictures and would work well indoors (low light) and outdoors. It also has to be user friendly, with an automatic AF feature. I'd like it to have an HD movie feature as well in addition to a strong zooming capacity. I had been debating whether I should move into the DSLR market, but I am not sure which camera to get. I had always been using the Sony brand and I am not sure if I should switch to a cannon at this point.
Any recommendations would be greatly appreciated.