The tools I use to clean my cast iron are a stiff brush on a handle and a washcloth shaped piece of chainmail that is made specifically to clean cast iron. Sometimes I use a green scrubby, but not very often.
To clean cast iron you just rinse with hot water and scrub. Never put it in the dishwasher. The first time I made caramel popcorn I boiled the honey and coconut sugar in my 2 quart dutch oven. I was certain that clean up would be impossible and I’d have to use soap and lose my seasoning. Instead I rinsed the pot with hot water for 20 or 30 seconds and it was perfectly clean.
Do not put hot cast iron under water. If the water is too cold, it will warp and ruin your pot. Just let it cool before you wash it. For caked on food, I put water in the pan and let it set for 10 minutes or so. I used to use green scrubbies to wipe out the soaked food. Now I use a stiff brush on a handle (my hands stay cleaner than if I was using a scrubbie.) However, for really tough spots I use a scrubber made out of chainmail. If the spots aren’t too tough, I use the brush with handle to move the chain mail around the pan to keep my hands dry.
Don’t let food just sit in your pan. One of the biggest problems I’ve had with cast iron rusting is leaving it on the stove with liquid or moist food in it overnight. The absolute biggest problem was when I did that and tried to clean up the rust using steel wool. The steel wool scratches the seasoning and I’ve lost huge areas of seasoning on a couple pans because of that. You can see the line where the original, thick, black seasoning meets the stripped, grayish areas. If something should stick, some people clean with coarse salt. I’ve never had a need for that, I just let it soak and use the chain mail. After you’ve washed it, dry it well (to avoid rust) and depending where you live, coat inside and out with a thin layer of oil. Since we’re in desert dry Tucson that’s not so critical, though I try and do it (well, sometimes I try and do it.) If you have a pot/pan with a lid, put a piece of folded paper towel between the lid and pan to make an air gap so that air can circulate.
If I do leave a pan overnight and it develops rust, I pour some vinegar on the rust, let it set, scrub it with a brush or chainmail, then wash it off. Keep in mind the acidic vinegar will eat at your seasoning, but it’s not as damaging as steel wool. I dry the pan, apply a thin layer of oil, and it’s usually fine. If you find an old pan that is heavily rusted, steel wool could help restore the pan. I just learned the hard way that steel wool can cause more harm than good if you want to keep the seasoning.
If you’re wanting to know how to season cast iron (and to see that cast iron can do just fine if its care isn’t perfect) see this page.