After a few solo flights, and a bunch of cancelled flights due to Canadian weather in the Fall, I've finally gotten in a dual. My instructor and I worked on Short Field take-offs and landings and Soft Field take-offs and landings in the circuit at CYKF. I only got to log 0.8 hours though, because we had to wait a while for the fuel truck. Also, the temperature as I was doing my walk-around was -4˚C out, and when I ran my hand across the wings, I noticed a light layer of frost, so I'll post about Winter Operations in a bit too.
First of all, as the name implies, we would use short field techniques when we have very small amounts of runway length. Pilot decision making applies here, because you would have to decide whether or not you needed some form of short field technique.
To perform a short take-off, you have to taxi to the very beginning of the runway, don't follow the taxiway centerline. Depending on traffic conditions, you may have to get ATC clearance to practice the short field take-off. After you taxi into position, hold your brakes firmly. Increase your throttle to full, and wait until your instruments are reading maximum power. After receiving the indication of full power, release the brakes and proceed to take-off.
To perform a short field landing, you will need to modify your approach from your normal landing. You will need to establish a much shallower approach slope, and you will need to use power to maintain altitude until the runway. When you get near the runway, begin your flare before the threshold, so that your touchdown point is as close to the threshold as possible. After that, in the DA20, you will raise all flaps (so that less lift is produced, so that more weight is on the wheels, which increases braking effectiveness), apply brakes, and hold rearward stick pressure (so that the elevators produce drag, which will aid in slowing the aircraft).
Typically, when practicing this technique in the circuit, you will have to request a stop-and-go, so that you can fully practice the take-off and landing technique.
Again, as the name applies, we will use the soft field technique when operating from non-paved airstrips, such as grass and gravel. It's a good idea to learn these with an instructor on a paved runway before going to a soft runway, to avoid damaging your airplane.
To perform to soft field take-off, you want to hold back pressure on the stick while gradually increasing power (make sure that your taxi and take-off are in a continuous movement so that your wheels don't become stuck). Since you are applying back pressure, make sure the tail does not hit the surface. Your aircraft will lift off earlier than normal, and after you gain a few feet of altitude, level off and allow your plane to accelerate in ground effect. If you don't let this happen, you aircraft will likely stall after exiting ground effect. After you reach your normal climb speed in ground effect, you can then continue your climb.
To perform the soft field landing, you will not have to modify your approach at all. Try to touch down in a nose up attitude (slightly higher than your normal landing), and use your elevators to keep balancing on the main wheels only, so that your nose wheel remains protected from the rough ground.
Generally, you will practice the soft field technique by doing touch-and-goes. This is relatively challenging because you will have to balance your aircraft of the main wheels throughout it's time on the ground, in addition to raising flaps, increasing power, maintaining directional control, ect.
I have another flight where I will practice these techniques solo, and then I will head to the practice area with my instructor for Precautionary Landings, Forced Landings and Diversions, and then after that I begin pilot navigation :D
For more info on these techniques, reference your Pilot Training Manual, your instructor, and I also recommend checking out some YouTube videos, to see visual demonstrations of these techniques.