Are you fully understanding the relationship between shutter speeds and lens apertures (and ISO)? Sorry if you've got this, but they are two halves of the same coin.
The sensor needs a certain amount of light for optimum exposure (though you can vary it with ISO, at the cost of increased noise). Shutter speeds are set in an obviously doubling/halving relationship, eg 1/125sec, 1/250sec etc, and these dove-tail with the doubling/halving of lens f/numbers.
It's not obvious, but f/numbers are linked by the square root of 2, ie 1.414. So, for example f/2.8 lets in twice as much light as f/4 (2.8 x 1.414 = 4), which is twice f/5.6, and so on. You can mix and match these settings with shutter speed to maintain correct exposure, but modifying the effect of shutter speed on movement and f/number on depth of field.
For example, 1/500sec at f/4 delivers the same amount of light (exposure) to the sensor as 1/60sec at f/11. You would use the former settings to freeze action, and the second set would give a lot of depth of field, depending on the subject and the effect you want.
The third corner of the triangle is ISO, which is also linked in the same doubling/halving way. So taking the example above again, if you were shooting at ISO200 but wanted to use an even higher shutter speed for fast action, raising the ISO to 400 would allow you set 1/1000sec at f/4 and still get correct exposure.
Finally, when changing shutter speeds be aware that camera movement (camera shake) is a big problem at longer settings. Camera shake is related to lens magnification so the longer the lens the higher the shutter speed must be to minimise shake. Rule of thumb is the focal length multiplied by the crop factor, eg if you are shooting at 60mm focal length, then the longest shutter speed you should use for hand-holding is 60 x 1.6 = 96.
1.6x is the Canon crop factor (NIkon/Sony is 1.5x) so in this case you would set 1/100sec or faster. Image Stabilisation helps a lot with camera shake, allowing you to use much longer shutter speeds than the rule of thumbs says, but it cannot do anything about subject movement.