The closest thing I have to a simple "point-and-shoot" (or "cheap") camera is my iPhone 4s (8 megapixels--compared to my T3i's 17.9 megapixels). While I'm not about to go do a professional photoshoot with my iPhone, I do use it on the go, and here are a few things that I have learned.
Understand Your Camera's Limitations
Don't expect a small camera to take the images that a larger, more expensive camera will take. There will be some areas where you'll need to compensate for your camera's lack of technology. For example, cheap cameras need good lighting to make clear images. You cannot take pictures "on the move" as easily (move "travel pictures" will be blurry). Also consider your camera's focal length -- some cameras can get within two inches of a flower, other cameras need about six inches to focus. Be willing to work with your camera. Make sure there's good lighting. Make sure your camera is focusing at the distance you have it.
Work on Your Eye
A camera should not limit your creativity. I have seen some pretty amazing pictures by people who have small cameras. Work on your angles. The tips that I mentioned in this post and this post can be applied to any camera. Get out, take pictures.
1. Keep your camera steady
Most small cameras don't like shaky hands. Give yourself a sturdy grip on the camera--use something external to anchor your arms/hands (for example, prop your elbows on the ground).
2. Have good lighting
Your camera will probably give you bad grain if you don't have good enough lighting. So, if you're going out to get some prize pictures, make sure you have good lighting.
3. Work on your angles
Don't let a small camera thwart your creativity! Work on your eye as you use your small camera.
From my Camera Bag (or card ;))
To be fair, here are some completely unedited pictures from my iPhone 4s. If you notice, with all of these, there is good lighting. And I'll also acknowledge that I was picky with what pictures I chose. Some pictures from my iPhone are grainy, "just for the memory's sake" pictures.
If you're still not convinced about a "small camera," here are some pictures that I took for my sister's business--using my iPhone. These, I did edit a little. However, two important elements in these pictures are: 1) Good lighting (it was midday), 2) Cute props.
What kind of camera do you use?
If you have a small camera, what are some tips you'd add here?