I would echo most of the comments already made.
Firstly, given the fact that they were taken on a compact camera (particularly one with which the user is as yet unfamiliar) there is little in these shots which it would be reasonable to take exception to. Assuming that you are not intending to make enormous prints then once you have taken on board a couple of technical matters you should not have too many problems (If you want huge prints you have possibly chosen the wrong camera anyway).
As has also been said there is an issue with shallow depth of field. Most of the shots (unless I am much mistaken) have been taken with the lens at its maximum aperture of f/2.8. Use aperture priority and stop the lens down as far as practicable.
Another issue to consider is whereabouts in the zoom range these have been taken. There is an inevitable loss of sharpness with any zoom lens as you move away from its midpoint and it is possible that this may have been a factor here. Zoom lenses have a tendency to make us lazy, it is often better to leave those buttons alone and make use of one's feet when composing a shot!
I can understand your desire to be able to simply point and shoot. Unfortunately in order to do so you may have to slightly lower your expectations.
Finally you need to perfect your technique in holding the camera in order to eliminate the slightest possibility of camera shake. The main principle is that your left hand takes almost all the weight of the camera leaving the right hand relaxed so that the index finger can gently squeeze the shutter release. Resist the temptation to pull your finger away after you have taken the shot. The lack of an optical viewfinder may become something of an issue here as it is far easier to hold a camera steady with it to your eye than when looking into a screen.
Can someone quickly give me a dumbed down version of this F mlarky please
The smaller the hole through which the light enters the lens the greater the area (front to back) that is in focus in the resulting image. To make things confusing the "f" number is a ratio so the bigger the number, the smaller the hole.
With your lens "wide open" at f2.8 there will not be much in focus in front of and behind the point on which you tell the camera to autofocus. By making the hole smaller - selecting f/5 or f/8 you will increase that range that is in focus.
The downside of this is that with a smaller aperture, less light gets in so the shutter needs to be open for longer to let in enough light for a correct exposure. If the shutter is open for a very long time it becomes impossible to hold the camera still so you get camera shake. You can compensate for this by increasing the ISO but this can also lead to issues with "noise" in the final image. (For film photography you have to choose the right ISO film to load - high ISO is more sensitive but also more grainy - at least you have the option of setting it differently for each shot as required)
Once you get your head around this interplay between aperture, shutter speed, depth of field (AKA depth of focus, AKA zone of sharpness) and ISO you will have an understanding of the most important techical aspect of photography which will stand you in good stead whatever camera you use.