ORIGINAL: HighPlains
Fly straight and level.
Fly a pattern in both directions
Land
Take Off
This is very typical and I think oh so wrong based on my experience. Takeoff and landing involves getting very close to the ground where mistakes hurt. They are also where the student pucker value gets high, so I avoid these subjects until after we spend a lot of time doing basic acrobatic flight.
The value of doing acro is two fold. First, it's what most of them want to do so it really gets their interest. Second, it places the airplane into unusal attitudes where they learn how to recover on their own. After they know how to recover from various combinations of loops, rolls, and stalls from any aspect (direction of flight), they have very little stress in flying a basic landing pattern, and tend not to be concerned with the left/right thing or overrolling into the ground. Toss in slow flight and flairing at the proper moment and landings are easy. As to takeoff or landing first, it really depends on the student.
Personally I believe my best skills were learned at the end of a simulator session, when I decided it was just time to fool around and do crazy maneuvers (I tried very hard to fly a perfect box pattern, figure 8's, etc until I grew tired) and recovering the plane from unexpected attitudes and positions really helped me a lot. For example, one of my first (live) take offs, off the buddy cord, got incredibly squirrely - I think I over corrected just before I left the ground, and wasn't giving it enough power. Anyway, the plane stalled in every direction, twisting and turning. My instructor just said to stick with it, and I did, and the plane never went past the lateral edges of the strip, nor crashed into the ground, and I was quickly in full control of the aircraft. That definitely wasn't learned trying to fly a basic pattern.
Also, while I was still on the cord I asked if I could do some 'crazy #$%^' with the plane. My instructor said sure, but made sure I was plenty high in the sky. Again, I reacted to the plane stalling, rolling, etc. and feel that this is where I gained a better understanding of the sticks.
After recovering from these two incidents, flying a perfect pattern (right side up and inverted) is a piece of cake.
Last thing, which I can't remember if was posted or not, is taxiing. That was the first thing I did - taxi up and down the runway, slowly at first, and increasing throttle. I got to the point where the plane was ready to fly but not airborne, then reduced throttle. Controlling the plane on the ground with enough speed to take off, IMO, is one of the hardest parts of the hobby. This past week there were 3 beginners that all crashed due to lack of control just before the plane left the ground (they managed to get the plane airborne but so badly out of control they couldn't recover). Here, over steering becomes very easy to do, and there is a transition, difficult to predict/understand for a beginner, between the front wheel and the rudder doing the job of keeping the plane straight, which is dependent on a few things, including elevator, wind, etc. What I didn't see any of those 3 pilots last week do is taxi at high speeds up and down the runway. They did some slower taxing around, making turns, etc. but nothing just short of flying speed.