Most of the indoor rides are pretty dark, and it would be hopeless without the right gear.
That means a dSLR/mirrorless with a fast lens (aperture no worse than 2.8, though 1.4 or 1.8 would be much better). Or an advanced point and shoot with such a lens.
Without that basic requirement, you * might* still get some decent shots of slow moving bright indoor attractions -- truthfully, "it's a Small World" is the only example that comes to mind.
So even for Small World -- you want to use the largest possible aperture ( smallest number). Since most lenses have variable aperture -- you will want to be zoomed all the way out to get the best aperture.
Manually crank up your ISO. I hate to go all the way up to 6400 (or higher), but it can be necessary for the indoor rides. I try to get by with 3200, but it can be too slow.
Finally, watch your shutter speed -- since the ride is moving, and often the attraction has moving parts as well, you need a pretty fast shutter speed to avoid blur. On auto, the camera will try to use a slow shutter speed to counter the low light. (So you can't use auto mode). I'd try to keep shutter speed to 1/50 or faster.