FWIW... I have never used sports mode, even though it exists on some of my gear. I predominantly use Av mode, to ensure I control the DoF, but keep an eye on the shutter speed, perhaps using auto ISO. If the speed element is significant, then I will use Tv mode, dial in the speed I want and then watch the aperture, again perhaps with auto ISO. It's always worked for me, but I certainly accept that these things are personal.
This takes some practise and experience to assess the DoF and SS desirable or acceptable for specific situations, but that comes with time.
This example is one of the trickier ones: I wanted to freeze the motion of the aircraft in the air, but keep movement of the propeller. I chose matrix metering as the aircraft were moving quickly across my line of sight, which worked well with the tracking system of my Canon 7DMkII, and the EF 100-400MkII lens. With plenty of light, I set the camera to Tv mode (since that was the most critical) @ 1/500sec, had the FL@ 400mm and f/10 (so I had a good DoF) and the ISO at 200.
On the other hand, at a baseball game (which is a more likely scenario for you), I chose to set my camera up differently. Again, the light was reasonable. I was using a Canon EOS 60D with the 70-300L lens.
I was using spot metering and focus (both back-button), and I decided aperture was more critical: so, in Av mode, I set that at f/7.1 (balancing a reasonable DoF with SS), ISO was still at 200 (it's my go-to setting, too many years shooting Ektachrome 200!). With those settings, I got a SS of 1/200sec which captured the movement of the extreme end of the bat, and caught the movement of the ball as it was struck by the bat, yet froze the people.
In the last scenario, I was using a Canon 650D (you would say a Rebel T4i) with the EF 24-105 f/4 lens. I had been using Av mode and kept it that way. Again, I was getting good light. I had the aperture at f/7.1 for some separation from the background, and ISO was 200, I had a SS of 1/250sec, which seemed reasonable - the trooper was not moving but I wanted the muzzle blast and that needed to be just long enough to allow for it to extend. I was using single-shot mode. I watched the loading and shooting drill once and then on the second round I took my shot.
Now, all of these photo's have one thing in common: they all had reasonable light. I don't shoot indoor sports, for example, so the experience of others would be very valuable in that context. What I do is decide beforehand what is critical to my expression of the image: do I need to freeze action, show movement (which suggest Tv), isolate the subject from the background or include it (Av)?
