Apologies in advance for the length of this post, but I believe an explanation would add credibility.
This one is really basic, and best of all, costs nothing (And it applies to camcorders too!) . . .
When shooting with a LCD equiped digital camera, avoid the temptation to hold the camera out at arm's length and aim/shoot with the LCD. The best pictures are taken shooting as with an ol' SLR, with the camera held to your face and looking through the viewfinder. Otherwise, you're greatly increasing your chances of blurry pictures. EXAMPLE: did a job at a convention where I was to take pictures of the convention's events provided by attendees and incorporate them into a PowerPoint for the final ceremony. I was given card after card where 75% of the pictures were unusable, simply because they were shot arms-extended and were too blurry.
It's a simple matter of physics and geometry . . . with your arms extended, it's so much harder to hold the camera still as you punch the button (excaberated by the delay some cameras have between button-push and the pic being recorded). By having your arms out in front of you with your elbows at an extreme angle (nearly 180 degrees or flat), you've lost almost all your leverage and ability to brace the camera, and any movement, no matter how slight (such as your finger pushing the button), is hugely amplified. Also, much easier on your arm muscles . . . how tired are you after videotaping Junior's soccer game holding your camcorder 2 or more feet out in front of your face for a couple of hours?
By holding the camera to your face, your arms are doubled up with your elbow angle nearly 0 degrees, greatly increasing stability, which can be enhanced by tucking your upper arms against your body. Also, either holding the camera physically in contact with your face, or placing the thumb of the hand holding the camera against your face, makes for rock-solid shooting.
Granted, you may have to compensate a teeny bit because the typical viewfinder is offset from the lens, but hardly an issue as digital pictures can be so large and then cropped.
You can compromise by holding the camera just a few inches in front of your face . . . not nearly as good, but much better and arms extended. Best of all is to follow all the tri-pod suggestions, after all it's how the pros do it.
There's also some techniques in how you actually hold the camera in your hands, but not as applicable to the smaller digi-cams these days.
Agree with those saying don't get caught up in the bigger-is-better game with regard to pixel-counts. Unless you're shooting professionally and/or need large blowups, middle of the road (4-5Mps) will yield fabulous shots. I even read an article theorizing that camera pixel-counts are getting close to exceeding what the human eye can detect.
Lastly, there's an expression in the biz, "It's all about the glass, Baby!" Which means lens quality matters. EXAMPLE: My 4MP Minolta has a typical consumer lens and shoots pretty good pictures. My Panasonic camcorder has a 4MP still camera built into it and provides noticably better pictures, all because of its superior Leica lens.