When doing any exercise, one must consider what muscle groups are being worked. This knowledge will allow you to perform exercises with better form and more effectiveness. Putting together various exercises into a workout needs thought and planning to maximize time spent working toward your goal.
The pushup is largely an upper body exercise and is also a compound exercise, which means that many muscle groups are utilized at once. The primary muscle worked during a pushup is the chest or pectoral muscle with the secondary muscles being the shoulders and the triceps (muscle on the backside of the upper arm). Diamond pushups will partially isolate the triceps and lead to more work being done by them. Handstand pushups focus tremendously on the shoulders and are a great way to strengthen them. Another way to think of handstand pushups is an inverted military press. If you do not know what that is do not worry about it, I will cover that exercise later, if you do know it, you get the idea that it is pretty intense; a military press using your own body weight is a feat of pure strength.
One thing to watch for when doing any compound exercise and especially the pushup is the muscle that hurts the most or fails the most quickly. This indicates the muscle group that is the weakest and also means that the other muscles are not being worked to their full potential. A good idea then, is to focus on getting the weak muscle stronger so that compound exercises work each muscle approximately with the same intensity. This is the ideal and most effective way to utilize these types of exercises and is something to pay attention to and adjust appropriately.
During a pushup, the core is also worked. Your abs and lower back should be tightening to keep proper form and will then be indirectly worked. Focusing on tightening your abs and lower back will aid in using proper form and making sure they are contracted will work them harder and you will get a better overall workout. Different forms of pushups will work your core with varying intensity; doing pushups with your hands on two balls will require great stabilization, which is one of the primary functions of the core, and will consequently intensify the work done by those muscles.