I grew up in a house with a similar problem, and thus learned all about how to avoid toilet water on the floor.
What you might want to try is getting somewhat familiar with the insides of the tank. The standard toilet design has a large floater (looks like a big rubber ball) on the end of a long rod. This part of the plumbing controls when the water flow stops. Once the water in the tank gets to a certain level, it pushes the floater and rod up, cutting off the water. The fastest way to avoid toilet water on your floor is by taking the lid off the tank and lifting the rod. We actually rigged up a little hook out of paperclips and a small chain that hooked onto the back of the tank and we could set the rod in there to keep it held up, because as soon as you let it go, the water's gonna start again. This method can be a LOT faster than fiddling around behind the toilet for the water cutoff valve, not to mention that if you're too slow turning the water off, you're likely to get a whole lot messier when the thing overflows.
Of course, the above method requires that you get really good at taking the lid off the tank, putting it on the sink or something, and lifting that rod, so get some practice in. Make sure that you don't keep a whole lot of stuff on the back of the toilet. You don't want to rush through everything and accidentally drop the toilet tank lid, as it's likely to just shatter if it hits the floor hard.
Once you've got the immediate problem solved, you've just got to plunge. If the plunger you're using is one of the regular red ones that just looks like a half-sphere, get a different one. I find that the ones that look like this:
Work a whole lot better. I'm guessing that the little thing coming off the end really helps push water down the pipe or something.