The first tip, is that when dying your hair, apply a border of Vaseline around the edge of your scalp. Make it an area perhaps an inch wide or more. Yes, it feels a little greasy, but it will save you the work of trying to scrub off dye stains at the base of your neck where you can’t even see what you’re doing.
If you do have hair dye stains, there are commercial removers to be bought at hair salons (after all, they make mistakes too), but you can get the same results with your toothpaste. The make up of toothpaste means that it contains just enough abrasive to help remove the dye from your skin, without irritating it. Rub it in with your gloved fingers, or use an old toothbrush.
When the dye falls on clothing, get it off and into cold water, immediately. Soak until the stain is all or mostly removed. Wring out and place on a firm surface, then use an old toothbrush to work a dishwashing detergent into the stain. Rinse and launder as usual but air dry in case the stain is not completely gone. If there is some mark left, soaking it overnight in one of the OxyClean products may help.