Welcome to the forum.
I think it's time for a basic photography lesson.
There are two main factors that effect the exposure of a photograph. Shutter speed (how long it's open) and aperture (the size of the opening). The shutter speed is in fractions of a second, as you know. The aperture is represented in a ratio called F-stop. (ratio between size of the opening & the focal length). The lower F number, the bigger the opening. (F 2.8 is bigger than F8 )
The two values are reciprical...if you double the size of the aperture and cut the exposure time by 1/2...you will get the same exposure. and visa versa.
So when you put the camera into shutter priority and set a fast shutter speed...the aperture has to open up to allow more light into the film/sensor. The problem is that aperture size is limited by the design of the lens. So if your biggest aperture is F4 and your camera's light meter is telling you to use F4 @ 1/125...and you want to shoot at 1/500...the aperture would have to be F2 (four times bigger) but your camera can't open that much...so you end up two stops underexposed. The faster you set the shutter, the more underexposed your photo will be.
You were able to use faster shutter speeds outside because there was more light.
To be able to get proper exposure with fast shutter speeds, you would need to use a lens (or camera) with a bigger maximum aperture or use faster film (bump up the ISO).
Now if you are close enough to use a flash, you can freeze the action with a much slower shutter speed because the flash burst is much, much faster than the shutter. At 1/125 with flash, you would actually have two images. One would be blurry from the ambient light and one would be sharp from the flash.
Hope some of this was helpful.