I know a part of biking endurance and speed come from having a perfect pedal stroke. What should I do to improve my pedal stroke?
Pushing and pulling and rhythm:
You'll certainly need clipless pedals - you won't develop a decent stroke if you're only pushing down on the pedal. You need to be pulling as well.
I would advocate spending some time riding a fixie, too. If you have to keep you legs moving, you will start to feel more connected to your bike and its speed.
Get yourself a cadence monitor and try to maintain an even cadence when in different gears. When you're not on your fixed, concentrate on spinning your legs. Perhaps try a gear or two higher than you might, purely to spin up a higher cadence. Also maybe trying a stationary spinning bike.
Pushing down on the pedals is a natural thing. So to improve the "roundness" of your pedal stroke, completely ignore the down-stroke, instead focus on the following:
Continuing the pedal stroke "across the bottom". This will probably feel something like trying to scrape mud off the bottom of your shoe.
On the up-stroke, focus on driving your knee upwards toward the handlebar.
At the top of the up-stroke, focus on sliding your foot over and down into the down-stroke.
Focusing on the non-instinctive 3/4 of the pedal stroke has really helped me improve form. Using a fixed gear (or staying in a low-ish gear), find someplace where you can spin out your cadence (a slight down-hill but not too steep).
Focusing on the 3/4 of the stroke that isn't pushing straight down, pedal as fast as you can until you find yourself starting to bounce. Keep doing this, and as you get better at the non-down-stroke, you'll find that the pedaling cadence at which you start bouncing will get higher and higher, as your pedal stroke gets more round.
One of the best reasons for having a round pedal stroke is efficiency. If you pedal stroke is smooth and efficient, you will get more power to the wheels with less effort. Some exercises for improving your pedal stroke; the winter is a great time for such low intensity exercises.
Pedal a very big gear up hill (20-40 rpm) for intervals of 5-10 min x 2-5/day
Pedal with one leg at a time 2-5x4-10 min x twice/week
Ride a fixed gear
combine two or more of the above.
Also note that it is not actually possible to get a perfect pedal stroke. I learned the above exercises from one of Massimo Testa's apprentices who was coaching the UC Davis cycling team. My pedal stroke was smoother than many professionals who had been through the lab, but as you can see (below) my dead spot (going over the top) was only 1/6 as powerful as my down stroke (at 90 degrees). This test was done with pressure sensors on each pedal. My right and left feet are shown by the blue and red lines respectively, at a constant power of 120W and about 90 rpm.
If you ride on a trainer some of the time, consider getting a set of rollers. They force you to ride smoothly at a high cadence so you don't bounce up and down, and I've seen them dramatically improve my girlfriend's pedalling technique. They're much better for your technique than a turbo trainer.