Traction control is on by default. If you hit the button for it, you turn it off (and a warning light comes on your dash showing TC off). If you hold down the button for a few seconds, you turn Stability Control off as well, and get a second light on.
Under normal conditions neither of these, nor the ABS, does anything at all. What they are doing is looking for your tires to spin, skid, or slip, and when they do, they activate an emergency response.
TC is looking for spin, when you try to accelerate and your tires lose grip with the road. When it detects spin, your ECU chokes the life out of your car, reducing you to about 20 horsepower for a few seconds, and providing a flashing warning light on your dash telling you this is happening. For many of us, this is worse than unnecessary, as we know how to respond to spin by throttling back just far enough to get traction, and for only as long as needed... which is probably a fraction of a second, not several seconds. YMMV, and if you disable TC you're assuming the risk, of course.
Stability Control is similar, it looks for you to lose traction in a turn, which is to say skidding. It is supposed to fix that by using your brakes in a differential fashion, braking some wheels while letting others turn normally... but there's no evidence the car is at all capable of doing that. What it seems to do, instead, is the same thing it does when TC goes off. Throttle you back and hope that helps. Again, this stays on for several seconds while the condition probably lasted a split second, at least if you're any good at driving. YMMV and so forth.
ABS, finally, is also similar, it looks for your tires to skid while you are braking. If this happens, it figures you are panic braking, and so it removes the brakes from your control and pumps them up and down rapidly. This is, in point of fact, the correct response to a skid... but as rendered in the Sonic, it is overly sensitive, the least tiny bit of skid will activate it, and it is astonishingly harsh, pounding the brakes on and off like someone took a jackhammer and mounted it to your shocks. Unlike the other two though, ABS can actually save your life by taking over braking in an emergency, where your reflexes might not do anything but stomp harder on the brake (which doesn't help at all). Of the three, I'd think hardest about killing this one... but many have, it's up to you.
These conditions can all happen in dry weather, but are more frequent in the wet, or mud, or snow or ice... anything that makes it harder for the tires to keep firm contact with the ground. Playing with the button (or even pulling the fuse) might make the car handle a little more the way you want it to under extreme conditions, but nothing you do there is going to matter in normal handling circumstances.