Modern cameras are marvels of technology when it comes to helping you take decent pictures with regards to exposure, but there are times when your camera can be "fooled" by difficult lighting situations, so sometimes you need to know when to tell the camera what to do rather than letting it run on automatic all the time. Also, there are a couple of basic rules of composition that will be helpful in getting photos that are more interesting to the viewer than your typical "point and shoot" operator captures. I'm no pro, but here are some things that will help capture those special moments. In no special order, and they are certainly not meant to be all inclusive.
1.) 99% of the time it's not the camera that takes a good picture, it's the operator. Todays pocket cameras are fully capable of taking a good photo if you use proper technique. So don't think it's your camera that's bad and somebody elses is "better". That's rarely the case.
2.) The Golden Rule of photography is called "The Rule of Thirds". Think of your viewfinder (or LCD screen) as a tic-tac-toe grid superimposed over what you're seeing (some cameras have a function to actually turn on a grid to make this easier). Place your subject, or in the case of a landscape photo, the horizon, at one of the intersections of the grid. In other words, don't bullseye your subject, with rare exception. Also, don't split the screen in half with the horizon, unless you're taking a photo of a reflection in a lake, pond, mirror, etc. If the sky is the more interesting thing in the photo, let it be 2/3 rds of the photo (sunsets). If it's the mountains or land, cut the sky down to 1/3 rd of the frame, at most.
3.) When taking a photo of a sweeping vista, it's generally better to have something (a tree, your bike, a bush...any object) in the foreground of the photo. This gives the photo a sense of depth.
4.) Move closer. Decide what about the scene you are looking at is the most important, or what drew your attention to the scene in the first place. Zoom in, or walk forward, to better frame that part of the scene. Capturing too much information is distracting and people can't figure out what it is you were trying to take a photo of.
5.) Learn how to turn your built in flash off, or on, rather than just leaving it on "Auto" all the time. Auto works great if you're taking photos of people inside, or outside, if you're close enough. 99% of the flashes on "point and shoot" cameras, or any camera that has a built in flash, will NOT light up anything that is more than 10 feet away from the front of the camera. Think about how many times you've seen people in football stadiums taking pictures...thousands of flashes going off at the same time. Not a single one of those flashes is doing anything other than using up the camera's battery, unless they're trying to take a photo of the person's head that's two rows in front of them. Turn the flash off, unless you are illuminating things 10-15 feet in front of you. Force the flash to be "on" (not in auto) when you want to illuminate a subject 10 feet or less in front of the camera, BUT THERE IS A STRONG LIGHT BEHIND THE SUBJECT. How may photos have you seen of people standing in front of the camera in the shade, and behind them is a brightly lit scene that bathed in sunshine? You can't see the people's faces because the camera's light meter is getting fooled by the background light, so it exposes the scene for the background rather than your intended subject, the people/person. Here's how you fix that problem: Put the flash "On" (force the flash to operate) and take the shot. Usually, there is a setting for the flash that enables you to reduce the amount of light output...play with that a little bit. This is called "Fill Flash"...in other words, the flash will kiss the people/person with just enough light to illuminate them, while still exposing the background properly. The goal here is to make the photo look like flash was not used at all...only you will know that it was. Most modern cameras even have a setting (you might have to dig in the menu) for "Fill Flash". Use it.
6.) A lot of cameras have an anti-vibration or image stabilization setting. Use it. It will help you get photos that would otherwise come out "blurry".
7.) If you want to freeze motion (like a bike passing by in front of the camera), you'll need to use a shutter speed of at least 1/250 th of a second. Faster is better, so 1/500 th + would be great.