Over time, I've found there are a handful of really simple things that anyone can do to take better pictures.
What are your favorites?
Understand the basics of aperture, shutter speed, and ISO.
Take your camera everywhere, including when you don't intend to use it.
Take a lot of pictures. If you find what you could perceive as an interesting subject or image, take many images varying angle, height, framing, working distance, aperture, focal length, etc.
When reviewing your own pictures, think about what you like and don't like for each one. What could be done to improve it (not just in post process, but when you were taking it, or even stuff like if that tree wasn't there, the clouds were better, a certain facial expression, etc).
Also look at photos and photogs that you like and try to understand why you like the work and how to do that.
Submit your good ones to critique forums and websites.
Critique other people's photos, it will help you see and take better pictures.
Participate in photo challenges. Once you spend a week agonizing over a good concept for a theme, you can look at what other people did. The pains of your own creative process combined with great ideas that others have executed will give you new insight.
Forget about gear.
For a day, limit yourself to one focal length.
Limit yourself to a certain number of pictures (once you have started to develop a photographic eye from taking and looking at many pictures), it will teach you to be careful and deliberate about creating a photograph.
"check your edges" -- think about everything that's inside the frame, and make sure you want it there. ("check your edges" meaning to look around the edges of the frame -- often things that straddle the edge of the frame are good candidates for either bringing further in (tops of heads, say, or feet sometimes) or excluding (random objects, a transition that doesn't need to be there, etc.)).