You don't want to put fabric softener on your towels, if you like soft towels. And you don't want to wash new clothes with other clothes in hot water.
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If you use bleach, it can only go on whites.
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Other than that, have at it.
Alternatively: wash like fabrics together. Things like denim, towels, even blankets go together, but combine with your other fabrics and it wears them down more quickly.
A piece of clothing can only ever mess up your laundry load on its first wash cycle since brand new clothing tends to bleed dye. And even then it might not happen when you wash it in cold water. However washing everything together will make the colors duller overtime. The whites and the blacks will become grey and the colors will just be dull. I personally separate blacks from everything because most of my clothing is black so I can just wash it all together to stop the color from fading so much. Obviously it's gonna fade in time no matter what you do but you can try to slow it down.
Also, fabric softeners don't work AND they're really bad for the environment, putting in a 1:1 mixture of water and vinegar (with maybe some drops of essential oils if you want it to smell good, don't worry it won't smell like vinegar anyway) will help preserve the colors better than softener.
Separating colors was more important when laundry soap/detergent was much harsher and clothes were dyed ineffectively. Now you can wash pretty much everything together in cold water and not have colors leeching. Your parents probably still separate their colors because that used to be the #1 rule of laundry, but you'll be fine if you don't.
,
If you use bleach, it can only go on whites.
,
Other than that, have at it.
Also, fabric softeners don't work AND they're really bad for the environment, putting in a 1:1 mixture of water and vinegar (with maybe some drops of essential oils if you want it to smell good, don't worry it won't smell like vinegar anyway) will help preserve the colors better than softener.