The photographic "rule of thirds" still applies to images taken on a camera phone. Try not to place your subject squarely in the middle of your frame, but a third of the way in. Think about the main focus of the shot.
I love symmetrical shots and sometimes these shots make the world look much more appealing. Look for natural symmetry, too, not just man-made.
Look at the light around you—where are the shadows? Move your subject or move yourself to get your shadow out of the shot or to get a better image. Don't use a flash at all if possible—work with the light you have.
Just taking a moment to look at the horizon or the angle of a sign can help you line up your shot so that it is perfectly straight. This, again, is a simple step that will make your photo look better. You can straighten your photo in the Instagram app itself but it's best to do the hard work first!
There are lots of lenses you can get for camera phones that can help you with close-up macro shots or give you a wider angle. I would recommend getting an Olloclip or the Sony QX10 lens that will easily connect to your camera phone and give you a better angle to work with.
There are lots of photo editing apps for iPhone or Android phones to alter your image before you put it on Instagram. My favorites are Snapseed (free) VSCO (free) and Camera+ ($1.99).
Look at the viewpoint of the photograph you are about to take—why don't you take the photo from up high or get on your knees to take a better and different view? Have a look at things #fromtheground.
Think about using tilt shift to affect your depth of field on your image. By selectively focusing on a subject in your photos you can bring them to life. Use the tilt shift button on the Instagram app or use an app like Tiltshift ($1.99) or TiltShift Generator ($0.99).
Selfies. If you really want to show the world what you look like today or where you are in the world at least take a good selfie. Don't hold your phone at arm's length. Use a mirror or a tripod with a self-timer (Camera+ has a great self-timer for up to 30 seconds). Try going further away from the camera and get a better shot. Also, soft lighting will help make you (and the picture) look much better!
Look at your camera phone and find out if there are any tips to take a better shot with that specific model. Use the highest resolution possible on your camera phone and look after your camera—scratches on the lens will ruin your photographs.