Michelle - you're getting there!
Short answer to your question, depending on which mode your camera is in, yes. If you're shooting totally manual, where you choose your aperture and shutter speed, then it won't unless you change it. If you are shooting in aperture priority, where YOU choose your aperture and the camera chooses the shutter, then yes, it will. Or the aperture will change if you change shutter speed, if you are shooting on shutter priority.
Y'all please correct me if I'm wrong here...
Long answer - you will find there will be several "correct" exposure combinations (shutter+aperture+ISO) for a shot, but you will get a different result with each.
The bigger your apeture is (small number) and the more light it is letting in, the camera will compensate and the shutter speed MAY be shorter (again, depends on your shooting mode). I may be wrong, and I didn't look at your exif information from your carousel shots, but it APPEARS that the second one has a slightly shorter shutter speed than the top one. It's irrelevant, really, but unless you've chosen both your shutter speed AND your aperture the camera will compensate by choosing one or the other. For each shutter speed change (for example from 1/125 to 1/250 sec) your aperture will change one stop. Let's say that you took a photo with an ISO of 400 at 1/250 sec with an aperture of 4 -if you change your shutter speed to 1/125 sec, then f-stop, or aperture, will change a stop. You are letting the light in longer since you slowed down your shutter speed, so unless the aperture changes (needs to be smaller with a longer shutter speed) you could have an overexposed photo.
The downside to a larger aperture (remember, larger aperture, smaller number) is a shallow depth of field and your focus will be more crucial, but a larger aperture will let in more light in low light situations.
Third side to this exposure triangle is your ISO. If your camera will shoot quality shots with a high ISO, it will open up your limitations with your shutter speed and aperture. They're all related, and they all work together to get a correct exposure. This will come into play when you're trying to shoot in existing light conditions, especially if you're trying to capture action, such as the barrel racing. you turn on the flash and all bets are off, so to speak. Livestock arena lighting STINKS, and frankly, for your first time shooting the event with such a new camera, you did a good job!