My suggestion is to try and be deliberate with your photography. Go on a short-ish walk around the block or through a park with the intention of taking a few photos (doesn't have to be a lot). When something catches your eye, pause and think about what you're trying to capture. Then take a few different photos—vary your position or angle, adjust the exposure up and down, etc. After you take the photos, maybe a day or two later, look at the pictures and evaluate what elements matched your intention and what didn't.
Try to practice this intentionality regularly, but not 100 percent of the time -- sometimes you just need to quickly snap a picture, that's OK.
Other tips:
Rule of thirds: Imagine your camera frame is split into a 3x3 grid (the iPhone camera can show you these, probably other digital cameras, too). Try to keep your subject along one of the imaginary lines or at the intersections
Make sure to look at the background, too. Try to avoid utility poles sticking out of someone's head, for example.
Zoom with your feet. Often, taking a few steps closer to your subject can make a huge difference.
edited to add:
I'm not a professional. I took a photography course in college 20 years ago where we used manual SLR cameras and developed our own black and white film. Now I pretty much only use my iPhone.
