Most people will at some point or another have the goal of doing more Push-ups, Sit-ups, and or Pull-ups/Dips.
Regardless of your fitness goals, improving your relative strength is a strong indicator of your overall fitness level. Wrestlers, Gymnast, Cheerleaders and other athletes who's main purpose is overcoming their own body weight without much other external resistance have tremendous levels of relative strength.
So, the question is how do you get to their level? Well, truth is some of their ability will be largely influenced by genetics and their anthropometrics (their respective body proportions).
However, all hope should NOT be lost!
There are many theories as to why these particular body weight exercises are so difficult. But, for the sake of this article we shall assume you are competent enough to know how to perform the movements safely and you have a relatively decent starting point (You can perform at least 1-3 reps of each).
Alright, So How Can You Get Better in 4-6 Weeks at these moevements?
Set Repetition Method.
Choose whichever exercise(s) you desire to improve, then perform one unbroken (No rest, or extended pauses) set for max reps. Now, take that number and multiply it by 2.5 to get your total rep target. If you're just getting started multiply that number by 1.8 instead to get a more realistic goal.
For example, let's say your max unbroken set of pull-ups is 10. Take 10 reps x 2.5 = 25, meaning 25 is your total rep target unless you use the 1.8x and get the goal of 18. So, your goal is to accumulate 18 or 25 strict pull-ups at least 3-4 times a week for the next 4-6 weeks.
The most efficient way to try and accumulate your target reps is to do them in your OTHER workouts throughout the week. For example, after a set of squats, crank out out 5 strict pull-ups during your rest breaks. Or, on a day you may be doing bench press you can get in extra sit up reps for your rest period.
The point is, you want to be practicing these movements as often as possible, so long as it does NOT IMPEDE the quality of your workout of the day. Regardless of how you get them done whether it is one giant set, or multiple mini-sets all that matters is that you accumulate your target reps of your chosen movement(s) before you leave the gym that day.
Hopefully before the 4-6 week mark, you should be able to reach your target rep range with ease. If you find that to be the case, do not hesitate to modify your moves to make them harder. One way of doing this is performing the moves with intentional eccentrics. Another way would be to add external resistance such as a weight vest. And finally you can perform the move more explosively.
Weighted Pushup (Added External)
Overcoming resistance with Speed
The set-repetition method may seem simple. Truthfully, it is all while being effective! Remember, this method is best suited for bodyweight exercises like pull-ups, push-ups, dips, handstand push-ups, muscle-ups, or other exercises that don't cause a lot of systemic fatigue. Trying to use this method with barbell exercises could be a recipe for disaster.
All in all, this is an excellent way to set goals and have measurable progress.