This is a long answer... :o)
Even if you don't want to get in contact with the technical details, there are some buttons on your camera that makes a lot of difference in casual shots. Those buttons allow you to do quite critical steps for every shot, let's go step by step in the order you should use them for each shot:
The zoom controls
These are usually positioned around the shutter button and are used to approximate your subject (zooming in) or to make it "go away" (zooming out). Usually the cameras offer stepped zoom controls, and it is useful to learn how to zoom in discrete steps instead of all the way in or out.
Use the zooming in to give more importance to certain subjects in the photo, like making a flower bigger than the surroundings. Use the zooming out to include more subjects in the photo, like a larger group of people or some landmark or landscape.
Once you find out how close or far you need your subject to appear in the shot, you can move to the next one:
The exposition control
This button (or option) is usually labeled like a +/- and allows you to inform the camera how light or dark you want your shot to be. Using it usually involves a ruler labeled from -2 to +2 and this numbers refer to the darkest setting (-2) and to the lightest setting (+2).
Usually leaving this control at zero lets the camera try to balance the photo in order to not make it too dark or light, but if you learn how to adjust the exposition of your shots, you certainly are going to get better results in most cases.
Also keep in mind that the camera is going to try to adjust itself to whatever subject you point and that this takes a while. Before adjusting the exposition, point your camera to the subject, wait until the camera "understands" the scene (usually around a second or so) and then proceed to focus and composing.
Darken your shots (move the exposition to a negative value) when you want to photograph something that is too bright. In the same way, lighten your shots (positive values) when you are pointing your camera to something dark.
Once you are happy with the appearance of your light conditions, move on to the final step:
The shutter button itself
Many people treat the shutter as just a place to press when you want to take a picture, but the shutter has to be handled gently and in most cameras (and some phones) it works in two steps. If you press it halfway it focus on the subject(s) and only when you continue to press it is that the shot is really taken.
Take a time to check if your camera have this two step action, learn how to reach the first step without going all the way and learn how to finish the pressing without shaking the camera.
The first half of the shutter action (fucusing) is extremely important, since if you don't give time for the camera to focus on whatever you are pointing it to, the shot will most probably come out blurred.
Another important use of the shutter half press is to allow you to focus on a subject and then, while still holding the shutter half way, move the camera around the subject in order to put it in a better position but still focused. This is related to the composition of the photo (see more about it below).
While not as much critical as the three above, most cameras also have other buttons that can help a lot taking better pictures, but you can take more time to learn then:
The flash control - usually represented by a lightning symbol. This allows you to turn on or off the flash itself. For most shots you probably won't need the flash, but try to shoot with and without the flash to learn how it impacts the shot. Leave the flash on if you are shooting something on a really dark environment, but understand that it will look too bright in the resulting photo. You should also turn it on if you are shooting some subject against the sun or other bright light source (this is called fill in flash).
The "mode" control - many cameras have "modes" that indicate how the camera will interpret what you are trying to do. Leaving this control in the "auto" setting will let the camera make all the important decisions for you and is probably enough for most casual shots, but some cameras don't allow you to control exposition or the flash in this mode. Another common modes are the "scene modes", where you set the control in order to inform that you want to shoot some scenery, sports event, people, close things (called macro) etc. These "scene modes" are very helpful and you should try them if you have the time.
The "macro" control is usually represented by a flower icon. This control let's you shoot something really close to the camera. Very useful for flower shots, but also for close details of objects, small animals or any other close up. Always remember to turn off the macro button before taking regular shots, otherwise your shots will come out really blurred.
So I just have to press the buttons in the right order?
No, but it really helps if you do. Once you understand what your camera controls do you certainly will start to be able to take better pictures. Think in the car controls you need to learn before going out in the street, if you don't know about the pedals or how to steer, you probably are going to have problems driving.
Once you know what happens when you press the buttons, it's time to start paying more attention to what you are showing in the pictures. That's the composition part of photography, and although not using the controls correctly are surely enough to get you a bad shot, using them correctly are not enough to get you a good one.
There are tons of books, sites and videos about composition and you should try to check some of this material if you have time. But this small list of hints may work as a starting point for shooting these subjects:
Landscapes - try to position yourself in order to include more parts of it and look out for things in the foreground (closer to you) that may be on the way.
People - try to position yourself (or the subject, if possible) in order to not have strong light coming from behind the subject or from directly above. Usually, if the subject is in the shade things are going to be easier for you and them (no squinting).
Action - be aware that simpler cameras are usually not adequate for action shots, but try to follow the subject as it moves. Not easy and not certain.
As you can see, positioning yourself or the camera is very important and usually neglected. Composition is much more about from where and how you point the camera and what is present in a scene than about what fits in the screen (which is more related to framing).
Also, keep in mind that having fun is a very important part of the exercise! I hope this anwser did not end being too much information. :o)