First off, it's best to understand what your camera is going to do on Auto. In this mode, all your camera can do is have a look at the light available and try to come with the correct exposure by setting the aperture, shutter speed and ISO for you. The problem is, the camera doesn't know what you are trying to photograph so it aims for a middle ground - middle of the road aperture, and middle of the road shutter speed - to try to cover all eventualities. This is why you will often end up with photos that have a "snapshot" quality to them - middle of the road settings is going to give you middle of the road photos most of the time. (OK, you might luck out from time to time but you know what I mean!) I have also noticed that cameras will also frequently give you a shutter speed that is far too low, resulting in blur in your shots - it's so common when shooting indoors. However, some very quick changes on your dial mode can improve your photos enormously, so here goes.
Scene modes are basically a way of telling the camera what you are photographing so that it can determine what is the most important setting, and it will work out an exposure based on that.
Perhaps you are photographing a running child where you want to freeze the action? Then choose "Sports" (or "Action") so your camera knows that shutter speed is important to it will choose a higher one to stop any motion blur. Or maybe you are photographing a snow scene where there is lots of lovely white snow which would normally trick your camera into underexposing? If you choose "Snow Scene" it knows not to try to dampen all that white to meet a middle grey so you will get a photo that is correctly exposed. Or perhaps you want to take a photo of someone and have a nice blurred background? Then choose "portrait", and it will assume that the most important setting is aperture, and it will select a lower one to help give you that blurred background. It's basically giving your camera a small clue as to what is going on so it can make some changes to the settings for you - it's not ideal or perfect, but it will get you a lot closer. Whilst I'm not a fan of staying on these too long, they can be of enormous help when you are starting out, as you can see the changes that the camera makes in the same lighting situation between each modes. Ultimately though, you are still more clever than your camera is even in Scene Mode, which is why you should to try to use....
These modes are like the halfway house between Auto and Manual, and the ones I recommend you explore (feel free to use the Scene modes whilst you are figuring it out!) These modes allow you to choose what your most important setting is, and the camera works out the correct exposure by setting the remaining two for you. The two main modes are Aperture and Shutter Priority. I pretty much always used Aperture Priority as I believe this gives you the most creative control for shooting children generally. Before I got more confident about the amount of light etc, I would start at around F4.5 and see what kind of shutter speed that would give me - if it was too low for what I was photographing I would simply lower my aperture number. If you are in a situation where a constant high shutter speed is vital - say a football match - then use Shutter Priority. In addition to helping you get better settings, using semi-automatic modes will also start to allow other settings in your camera to come into play - for example you can turn off your flash or shoot in RAW. Have a look at this recent post about choosing your camera settings to help you understand and choose which is your most important setting.
Manual with AUTO ISO
Once you are fairly confident in Aperture Priority and you have had a chance to have a play around, be brave and make the switch to Manual. If you still like the idea of the camera making the final adjustments for you, you can always shoot with AUTO ISO checked. This way, you can dial in your aperture and shutter speed settings, then the camera will choose an ISO setting for you that will allow create the correct exposure. It's great for when you are in situations where the light is constantly changing (or your subject is moving around a lot in and out of different light) as it helps keep the exposure constant without having to continually change your settings. I quite like to use this outdoors when Callum is on the go and frequently moves into different lighting, as I can follow him around without caring too much about changing settings as I know I have the aperture and shutter speed I need - the ISO is just the balance to get the correct exposure in this case. There is a downside to shooting like this in that you cannot decide to shoot "over" or "under", as the camera makes it so the little bar is in the middle, and exposure compensation doesn't work when you are in Manual. In cases where you need to do that (backlit images are one example) switch to full manual, or aperture priority with exposure compensation.
If you are still on AUTO, start to make some changes today, even if it is to just use some of the scene modes - I promise you will thank yourself for doing it! As usual, any questions, feel free to ask!