Self-taught and trained.
Recommendations:
1. Get an account on Flickr.com (yes, it is spelled "flickr.com" rather than "flicker.com") They have literally millions of pictures there contributed by every type of photographer you can imagine. Search for pictures that are like the ones you enjoy taking (i.e. landscapes, flowers, portraits, architecture, sports etc.) Sift through them until you see some that make you go "WOW! Look at that!" Then try and figure out how they took the picture, and go out and try and duplicate it.
2. Either take a night school course, or get someone to teach you the basics of how shutter and aperature interrelate, and how they affect depth of field, motion etc. Don't get bent out of shape trying to memorize all the numbers etc., just try and get your head around the concepts of what they do.
3. Composition is the key to a good photograph. Learn the "Rule of Thirds" and look for it in all types of photographs and graphic art. After awhile it becomes second nature. It doesn't matter how great the lighting is, or specatular the subject might be, if the picture is not composed properly or is not quite straight, the eye will notice it subliminally and it will seem "just not quite right."
4. Find someone beaming with enthusiasm about photography. See if you can tag along on one of their shoots. Watch everything they do. Pick their brains. Have them critique your pictures and maybe even re-crop them for you explaining why they would change it.
5. I think the best way to improve your eye is by watching a professional make what seems to you as being a slight adjustment, but to them makes a vast improvement on the picture. Those are the adjustments that demonstrate how my eye is not quite as well tuned as is the person's who is teaching me. Kind of like playing a piano that is just not quite in tune. To the beginner it sounds just fine, but to the professional the piano sounds just aweful.
6. Take several variations of the same picture of your subject. Move the camera back a bit and zoom in, move closer and zoom out. Move the camera a little higher or lower. Try changing that mysterious aperature setting and see if the background stays more or less in focus. Try different shutter speeds and see if you can freeze motion solid without underexposing the picture. When you look at them later, certain pictures will look better than others, and you'll find a commonality among the ones that look good. Find out what that is, and remember it. Look for it in professional photographs (i.e. magazines, movies, posters etc.)
Holy cow, I sure rambled. Sorry. Hope it helps.