It's quite tricky to solidly state that any amount of simple teaching will help people become incredible. As teachers of photography, we give our students an awful lot of work to get done. They go through a series of exercises that are mandatory before they come back for the next set of lessons.
When you learn anything, be it a language, how to draw, how to play cricket or any number of things for that matter (the one I find most annoying is dancing), we are essentially so engrossed in the trivialities of steps. The left leg - right leg stuff that makes dancing (or anything for that matter) terribly boring. Why couldn't we just take one class and then suddenly find ourselves become exceptional? Why could we simply not enjoy the bigger picture that we're so happy to see on television or at a show of some sort?
Sadly, learning is a something that requires a great deal of effort and development. In photography, we will suggest some very basic tricks that help most people get significantly better. I can't exactly think of the dancing equivalent but perhaps this is what makes photography so special. Unlike dancing, you have have been exercising your eyes and viewing images and compositions for some time now. Lets get down to the two simple rules that should help you take consistently better photographs.
1) Keep your composition simple
I can't say this loud enough. I often come across some photographs from beginners that have too busy of a composition. In fact, I would say that this is something rather ubiquitous among beginners. STOP! Take a step back and think about the situation and the photograph itself. Think back to becoming a viewer of the image you just took and ask yourself whether your eyes literally stay focused on the 'main subject' - that crucial subject that made you take the decision of taking the photograph in the first place.
Chances are that it is your perspective that needs tweaking. If you haven't quite managed to keep your composition simple, then go at it again. Simple tricks involve shallower depth of field and taking a different perspective (such as shooting at eye-level instead of from above.
2) Get Closer
Far too often, especially with portraits, I see rather shy photographs of people. They work some times but most of the time, they are simply just photographs that missed the mark. Nothing quite special about them. You need to get intimate with the photograph and for this you need to get closer. As a viewer of the image, I am quite interested in the subjects personality (assuming you are taking a portrait or a photograph of a group of people). Tell me the story with confidence. Not with some kind of shoddy reluctance. Get close and intimate so that the expression (smile, sadness, distracted) can be well communicated to me.
When you get closer, you also do the job of isolating things around the subject to make it the main focus. As long as you are careful to not chop off hands and legs, you should be quite alright here.
1) Simplify your composition
2) Get closer to your subject
If you can do these two things, you will probably find that your photographs would improve significantly and will capture something special about it all.
See if you can find such simplicity on the images on the side. Some are easier than others but you will see that even in areas of busy composition, we use shallow depth of field as well as exposure to simplify composition. The eyes always keep to our main subject.