Here are, in my opnion, the top ten bodyweight exercises you should be doing. It may seem odd to you that a seller of exercise sandbags is recommending bodyweight exercises, but we are. We mix them in with sanbag exercises, and sometimes weight the bodyweight exercises with a small sandbag.
1 PUSHUP is a good place to start. It is the most common bodyweight exercise, and for good reason. There are more ways you can do pushups than I can mention here. You can even do them in the office. Just do them with your feel on the floor, and your hands on the desk. Or, hands on the wall. Or, with one hand at a time. You can vary the resistance by varying the angle. My favorite way to do them is with my hands on basketballs and my feet on a small 7" tall stool. Also, I weight them sometimes with a small sandbag.I think pushups are better than bench presses, both in benefits and in safety.
2 PULLUP This to me is the best back exercise there is. If that is all it is, it would be fine. But there is more benefit to it. It is also one of the best arm exercises there is. Yes, it exercises the biceps. But it also exercises the forearm and the grip too. You can make it easier by standing on a stool, or you can make it harder and better for you by wearing some sort of weight such as a chain vest or a sandbag between your ankles. To take the pullup further, the ROPE CLIMB is the next level up. It is tough enough to work your whole upper body, but especially back, biceps, grip, and forearms. A popular exercise with CrossFit and other is the MUSCLE UP. This exercise is the extreme version of a pullup and a dip combined. It may be too difficult for many. If so, break it in two and do pullups and dips in the same workout.
3 DIP Back we go to chest and triceps exercises. This one really works the chest and triceps. I am not big on mentioning muscles by name, but I will keep it to a minimum. In my opinion, you should exercise MOVES, not muscles. Dips can be done either with bodyweight alone or with added weight.
4 SQUAT This is the exercise for the front of the legs, called the quads by bodybuilders. They can be done either with or without added weight.
5 LUNGES Great exercise for the backs of the legs, called the hamstrings.
Both squats and lunges are hard on your knees, and as you age can become painful. You can substitute the next exercise for them.
6 STAIR CLIMBING Can be done indoors, outdoors on stadiums, and even over a row of stools. Also, a variations is what I call the GIANT STEPOVER. I use an 18" high plyometric box and step up and over it with the same leg. Then do the other leg. This works both the front and backside of the leg all in one move.
7 SITUP Is getting over decades of bad rap. CrossFit has brought it back from the dead. The addition of some sort of padding under the lower back has been the trick that has saved the situp. This forces the lower back to keep its natural sway and not be stressed in unnatural ways. A pillow works well, or there is an inexpensive device called the AbMat that works well too.
8 BURPEES could very well be at or near the top of this list. This is because the move combines a squat and a pushup to work almost your entire body. Start from standing, drop down on your hands, and thrust your legs and feet behind you, and do a pushup. Then stand up and repeat. A great total body exercise.
9 SWIMMING This is what I do on vacation and also at home in the summer. It just is a great way to combine exercise and recovery all in one. When you are due to workout, and are still sore from the previous workout, your body is telling you something. You can get that workout and help recover from the previous workout by swimming.
10 RUNNING could be combined with walking in this list. The latest way to run is in intervals, mixing running and walking. That introduces recovery into the act.
There is much more functional fitness advice here.