Like bballdad and jrbrum said, push-ups are absolutely not a waste of time. They teach proper movement of your shoulder blade and work both your chest and tricep muscles.
In fact, there are numerous work-outs you can do without any equipment. Some of them being:
Military push-ups
Diamond push-ups
Sit-ups
Leg lifts
Bicycle crunches
Air squats
Lunges
Mountain climbers
Burpees
Planks
Jumping jacks
The list goes on, but these are just a few suggestions.
Also, if you can get your hands on a truck-sized inner tube, some duct tape, and some sand, you can build yourself a sort-of Bulgarian bag that you can use for additional weight in your lunges, squats, push ups, etc.