Know Your Goal
There are two competing attributes when it comes to physical training: strength/power vs. endurance. This is reflected in the very makeup of our muscles, which contain fast-twitch fibers for short, high-intensity movements, and slow-twitch fibers, which are weaker but can maintain movement for a much longer time. Also, fast-twitch fibers generally run on glycogen (an anaerobic process), which is created slowly and stored in your muscles and liver, while slow-twitch are able to burn fat and glucose aerobically.
You can train for either or both, but it's pretty difficult to reach peak condition for both (powerlifters are generally bad marathon runners, and vice versa). I'll give some brief tips for both, and you can do more digging on your own.
Aerobic
The most important muscle in your body is your heart. It's the only muscle that works 24/7 every day of your life. The best way to "train" this muscle is to create a high oxygen demand on your body, which forces your heart to pump more oxygenated blood. This is why you want to reach a moderate level of aerobic fitness, no matter what your ultimate goal is. While there are lots of good aerobic exercises, one of the best is simply running.
Running is obviously a body weight exercise, and requires no special equipment (although decent shoes help). You should aim for a continuous 2-mile run. If you can get up to 6 mph, this will be a 20-minute exercise. If you have trouble running that distance, start with a 1 mile goal, and run slowly enough that you can maintain the same speed over the whole distance. This might be barely over walking speed to start. That's ok. The point is that you want to train your heart and lungs to work harder for a long time, and train your mind to accept the idea of continuous load over a whole mile or two. Start easy and only run 2-3 times a week if you feel that's all your body can handle, and increase the frequency as you can, up to 5x a week. 4 mph is a very fast walk, so most folks can "jog" at 4.5 mph for a mile, even without training. Getting to 5 mph should be easy, and 6 mph should be achievable by almost anyone. If you can get to 7-8 mph, then you're doing well and should talk to serious runners about how to progress.
I would argue that running is a better warmup than jumping jacks, because while both put a load on your skeleton, which is good for your bones, running creates a much higher load because of the forward movement. Also, it's a lot less boring to run outside for 20m than doing jumping jacks. Finally, I only run 1 mile for warmup, and 2 miles for aerobic training. If you want to run races, you should train for longer distances, but if you just want basic aerobic fitness, 2 miles is a good distance.
Anaerobic
In order to change your body, you need to know this one simple trick: your body has a genetically programmed "baseline" muscle mass which it will always seek, and to increase your strength, you need to convince your body that it is not strong enough. This happens when the intensity of your workout causes micro-tears in your muscle fibers, which stimulates your body's repair mechanisms. The body has two rules: "If it ain't broke, don't fix it", and: "If it broke, fix it better than it was before." This is also why bones which break can become even stronger: the body assumes your bone broke because it was too weak, so the repair process makes it stronger. The same goes for your muscle.
The problem with push-ups and bodyweight squats is that you can quickly reach a level of strength which no longer challenges your muscles. If you want to get stronger, you need to provide more resistance. As others noted, you can do this by adding weights. Unfortunately, this becomes pretty challenging without proper equipment. If you're really hard-pressed for cash, you can make a squat bar out of a sturdy tree branch, some rope, and some water jugs, but I can't say I would recommend it. I would try deadlifts instead of squats, because you can improvise weights for deadlift easier than for squat (with a squat, the weight has to get up to your shoulders, but a decently fit lifter can easily squat more than their body weight, which is pretty hard to lift onto your shoulders safely and by yourself).
A push-up is just an upside-down bench press, so anything you can put on your upper back without slipping will add resistance. A bag of sand should do the trick, although positioning it would get pretty annoying.
Gym
If you can spring for a gym membership, even the cheapest gym with free weights will help your progress tremendously. With consistent effort, you should be able to bench press your body weight and squat 1.5x your body weight (in a year, most likely, but quite possible in less than 6 months). Note that a push-up is only about 60% of your body weight because you aren't lifting your legs, which contain a pretty significant amount of your total mass. At that point, I think you would be quite happy with your energy level and stamina.
If you can make it into a gym, you can train most of your muscles with just 4 exercises: bench press, pull-up, shoulder press/clean, and squat. A simple workout plan is called the 5x5: do 5 sets of 5 reps, with the highest weight such that you can just barely finish the last rep without a spotter. Start with a low weight that you know you can do, and work your way up until you find your limits. Ideally, you want to work your muscles to the fatigue point at each workout, because this is what causes them to develop most quickly. But without a spotter, this can be risky. So stay safe and always quit if there is a doubt that you can do another rep. Or, ask someone at the gym to spot you. For pull-ups, it is useful to use a machine that gives you assist until you are strong enough for your full body weight. Alternatively, you can use a "lat pull-down" machine to achieve a similar effect. Increase your strength until your pull-down weight equals your body weight, then switch to pull-ups.
Conclusion
The program I've sketched above will target strength more than anything. I think this is beneficial, because it should also increase your muscle mass, which will increase your base metabolic rate and make it easier to burn excess calories (the ultimate first world problem). Also, having excess strength makes a lot of things easier (and less injury-prone), and carrying around extra protein is actually protective if you get sick for a long time. If you are keeping up with the aerobic portion, you should also see your resting heart rate decline. This is a good indicator of heart health (a stronger heart uses fewer more powerful contractions to pump blood). Maintaining good cardio fitness is one of the keys to living longer.
If you find a sport that requires moderate endurance, then excess strength can provide that. You'll find that with higher strength, you can perform low-effort actions for a much longer time. In this way, strength confers endurance up to a point (for low-intensity activities). For instance, you should be able to do many more push-ups with a max bench of 180 lbs. than a max of 120 lbs., regardless of how much push-up training you've done. If you pursue a high-endurance sport, especially one with high effort, then you will likely have to change your training to fit that profile. But at that point, you will hopefully have learned enough to know what you want and how to get there. Good luck!
