Start with a regular push up
The goal is to do 3 sets of 12-15 reps. If it’s hard for you, then you can start by doing lower number of push ups. (8,9,10 for example). Go all the way to the ground and then explode all the way back up. Rest only 30 seconds between sets.
2nd exercise is the close hand push up
The goal here is to do 3 sets of 10 reps. Just like in the first exercise your rest periods between sets should be 30 seconds. Again, all the way to the ground and all the way up with explosive movement.
3rd exercise is elevated push ups
This exercise will be harder than the previous two, so the goal is to do 3 sets of 8 reps. You may also need more than 30 seconds of rest between sets so use 45 to 60 seconds of rest on this exercise. Go all the way to the ground and then explode all the way back up.
Remember, warm up well before starting the workout and stretch after the workout. Try to add 1-2 repetitions every workout, or 3-6 repetitions every week. In about 10-15 weeks of regular workouts you should be able to do 100 push ups in a single set.