Newbie to Sports Photography

Hello everyone, although I've been a member here for quite a few years and I haven't posted in a long time, I am still VERY new to photography.
I was fortunate enough to pick up a used Canon 5D MKIII. I've been trying to photograph my kids sporting events but I seem to be having a hard time focusing. Many of the shots I took focused on the background rather than the kids I was trying to shoot. I have a 24-70 2.8L and a 70-200 2.8L that I have tried using. Does anyone have any advice how to assist me with the focusing? As I mentioned, I bought the camera used and I'm trying to rule out user error before I think the camera is not working properly. The camera looks good and seems to work properly, it only has 41,000 shutter activations. Any help is appreciated!

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It almost certainly is user error.
While some lenses do front or back focus it wouldn't be by anything like that amount... so we can eliminate that. That leaves missed focus (as in you missed focus). That could be because your technique isn't quite good enough or because you are using the wrong settings.

Your settings should be AI servo, Single point AF, (settings on the lens - IS mode on and set to Mode 1). You should also check how far you are from the action as there are switches to alter the minimum focus distance. If you are over 2.5 meters from the subject select that, as it reduces the amount of hunting the lens will do when trying to acquire focus. Lastly go into the camera menu AF tab (AF1) and check that you case Case 2 selected.

Next up practice. Get a teddy bear or small light item and tie it on a length of string. Hang it from a door frame/light fitting/etc and set it swinging. Practice tracking it with the single AF point. Also watch what happens when the item swings away from the AF point (will focus on background) and practice re-acquiring focus by getting the AF point over the subject again.

Lastly you need to sort out your shutter/ISO/aperture. Based on the images you posted your shutter speed is too low, which will result in unwanted subject motion blur. Shutter should be a min of 1/500, if not 1/1000, f/2.8 and the ISO at whatever it needs to be to get a decent exposure. Indoor gyms are notorious for having bad lighting so you will have noise. It is just a fact of physics... do Noise Reduction in post to clean this up.

Besides the excellent post from Dan, you might consider the 85mm F/1.8 in the future.

While a lot of refs won't allow direct flash, nobody notices flash bounced off the high ceiling. Take a look at Curtis's excellent thread: No more excuses about high ceilings and bounced flash

Another consideration is to enable back-button-focus. I am shooting the same as you, except with a 7D MK II & a version 1 of the 70-200. I have found that having the ability to hold the back button to maintain focus as the subject moves to be much easier than the shutter button, I am not tying focus & exposure together. Seems to work well and those stinking gyms, yes, the lighting is horrible most of the time. I set the ISO to auto for most of my indoor gym shots as the lighting is all over the place.

Switch to Center point focus. If you have all of the focal points active, the camera can't decide which subject you want the focus to be on.

Excellent advice and information has already been posted. May I ask what settings you are using right now? More importantly AF settings. The 5D3 is a very capable, all around camera that does well in most sports and action.

Agree with the above. A shutter speed of 1/500 should be OK with MS basketball. I use 1/640 quite often for girls HS basketball but the lighting is likely a bit better.