Shutter speed should really not have that impact, can you post an example?

Close the aperture if you use a slow shutter speed.

As you're letting more light onto the sensor, so you need to compensate with the aperture.

The term you're looking for is over exposed. Either close the apperture (higher number) or lower the ISO.

It's a very basic concept, smaller hole needs longer to get light in etc, but its just something you have to go out and play around with in different conditions to get used to. Shooting in apperture and shutter priority is always good because you can look at what other settings the camera has chosen as a guide for yourself when using manual.