Sorting your laundry into colours and whites prior to washing is important and the colour fast of your items should always be checked. Generally speaking, coloured washing should be cleaned with a colour detergent as it does not contain bleach. This will help to protect the colour from being washed out of your clothes and other items.
For whites, bleach based detergent is needed to maintain the crisp white colour of your laundry, hence a white detergent should always be used for this reason.
For colourfast items where the dye is unlikely to run, this is less of a concern. However for non-colourfast items, caution should be taken and the correct detergent should always be used to minimise the risk of colours running.
Also look closely at the garment care label on your clothing and choose the right temperature and cycle for the load.