The choice of word might depend on the kind of toy, and who you're talking to. First, let's note that some grammar sticklers might say that you shouldn't use "fast" as an adverb and you should use "quickly" instead. It's such a common usage, though, that in everyday conversation they would be unreasonable to object, but in formal writing I might choose another word like "rapidly," "speedily," etc. Also, as others have noted, "across the floor" is more idiomatic than "on," and both could be left out if you wanted.
Now, what's the best verb? Is the car moving by itself, or is the child pushing it? Maybe it's a toy that can be wound up, like the kind you roll backward and then let go, or maybe it's a remote-control toy. For these, I would keep the structure we have here that makes "the car" the subject. As others have suggested, "rolling" is probably the simplest choice, and there are other colorful choices like "speeding" or "zooming." Of your suggestions, there's nothing wrong with "going" or "moving"; they're just unlikely choices because they're so vague. But "riding" would be a mistake because it usually means "being moved by something": a person rides in a car.
BUT: I'm going to suggest that you could say "the car is driving" if you're entering into the child's imaginative play. To him, the car is driving, and it's not inappropriate to speak about it using this imaginary frame of reference. "Wow! That car is driving so fast! I hope it can steer well! I wonder whether it has enough gas?"
Meanwhile, if the child is actually pushing the toy along, and if I'm not using this imaginary language, then I think I would make the child the subject of the sentence. I wouldn't say "the car is rolling across the floor" when it's being pushed; I would say "my son is pushing the car across the floor."
