Keeping your watch clean may be a bit more intuitive, but your watch straps may need a different type of care when maintaining it! Here are some general guides on how to clean different kinds of watch straps you may have.
Remove your strap, wipe it down with a towel or a soft brush, and place it in a small bowl that has warm water and soap in it. Soak your strap for a couple minutes and gently wash your strap in the bowl. Rinse it with running water, and gently press the water out of the strap between a folded hand towel. Hang the strap to dry it, or lay it in on a flat surface.
Rubber is a lot more durable, and are therefore easier to take care of. Remove your strap from your watch, and rinse it under warm water. You can use gentle soap to wash the strap down too. Afterwards, dry off the strap but keep it out of direct sunlight.
I hope this gave you a quick overview on how to clean and take care of your watch straps. Just as with regular clothes or accessories, make sure to treat your straps in a gentle manner and they will last you a while!