As a fitness coach, I train primarily with calisthenics, or body weight exercise, and walking/jogging/running. The only time you really need weights or special equipment is for bodybuilding. If you're just looking to get more fit, be more active, or lose weight- you don't have to spend a penny. (Although a nice pair of running shoes will make your feet happy.)
AnnaStephens already hit on the basics of calisthenic training- push-ups, pull-ups, planks, situps and squats. Of course there are many variations of each that target different individual muscles. Below is a good site that lists tons of exercises you can do... I know I've seen other sites that allow you to search for only the exercises that use no equipment as well.
Also, because you're exercising as a family, and there's a lot more benefit there than the fitness, I recommend simply playing. Any active game you can find to play. Not necessarily sports, although that would be great too, but even things like hide and go seek, or tag. Jumping rope is great too. Just think like a kid... we used to love just running around and wearing ourselves out- get back into that mindset. Be silly, have fun, laugh and play.
Good luck, and let me know if you have any more questions :)