There's been alot of good advice so far. I'll just offer a couple things that I think are important.
The push-up is a minor core exercise. Requiring abs,hip flexors,triceps,delts,erector spinae,lats, and the pec major/minor. When you see a bad Push-Up its because everything but the pecs has been neglected through poor form and discipline.
While the ballistic load is carried by the triceps and pectorals, the toe to nose is held stable by the core. I think its important understand that from the get go.
To develop increased strength and endurance you MUST EXECUTE PROPER FORM IN TRAINING, every rep. Although your form will degrade rapidly as you fatigue it should not degrade because your trying to go fast by dropping to the down position.
Proper Training Form
In the proper push-up position slowly, under complete control, lower yourself to the down position. The movement from UP to DOWN should take 2-3 seconds.
While in the DOWN position take .5-1 second pause.
Move from DOWN to UP in a slow and controlled manner. This should should take 2-3 seconds. Once you are at the apex of the movement contract your pec muscle completely for a second.
REPEAT
Training the Push-Up slow and disciplined to failure, repeatedly, will build stronger and more efficient muscles.
You can use Pyramids, you can do sets of a given number increasing weekly, Push-Ups to failure x4. You can google yourself silly looking for set and rep structure. Most anything done to failure, repeatedly will work.
What about speed for the PRT? You must increase strength and efficiency to improve your scores on any timed event. No matter what it is. The above will improve strength through technique and work.
Taking your stronger muscle to efficient muscles is all about oxygen efficiency.
Do your push-ups with the SAME TECHNIQUE as above.
Do an increased number of sets, lower reps per set. Decreasing rest between sets. This will force the muscle to do the work with less blood oxygen.
Like this: Ten sets, two reps per set. Two seconds rest between sets.
Or: Five sets or twelve with eight seconds rest between.
The numbers are obviously dependent on your ability but at the end of the sets you should be panting, sweating, and miserable.
Another way to starve muscles of O2 is by holding your breathe during sets. NEVER DO THIS RUNNING OR SWIMMING. Its known as hypoxic training and isnt for the novice. But you wont be a novice long.
The bottom line is WORK. You must work. You must harden yourself.
There have been what? 30 posts of advice on the Push-Up?
You will improve your push-ups through hard work. Really? If you told me you didnt care about technique or sets and all that sh*t? But beat the crap out of yourself every day? I'de be confident you will find success. But you can study the Push-Up like a monk, spend hours dissecting this thread and it will be a moot point if you dont get down and put in the time and pain required to harden the blade.
Relentless work ethic will beat any plan, any day. But a good plan in the hands of a relentless worker will beat anything, anytime.
So go do push-ups...