Rabbits are a good choice if you have a garden, as most leafy vegetable greens can be fed to the rabbits. With this approach, much of the unused part of the plant can be recycled as treats/food for the rabbit. This is also true for dandelions, clover, thistles, and many other weeds that tend to pop up in gardens. You'll need to do research on this because rabbit digestive systems are sensitive and feeding them the wrong food can sometimes be fatal even if the food is not poisonous. Avoid foods that are "gassy" foods like broccoli, as rabbits do not pass gas easily and this can be an uncomfortable, sometimes fatal condition.
You can create a worm farm with their droppings. The worms will speed the composting of the droppings and provide a harvest-able and sustainable crop of earthworms as well as providing an excellent top-soil/fertilizer for your garden. The droppings can also be applied directly to your garden as fertilizer as well.
Rabbit food is relatively inexpensive and a small herd can reproduce and grow fast enough to supplement a good portion the meat needs of a small family. It is important to choose the right breed for your climate. Some rabbits do not tolerate heat well at all. Most rabbits tolerate cold so long as their water does not freeze.
The American Rabbit Breeders Association(ARBA) is a group dedicated to the promotion of domestic rabbit breeding in the US. They have many good sources of information as well as local clubs that can be a resource. While the ARBA is primarily a show rabbit organization many breeders also raise for meat and can help you get started.