The Cleaning Rod
A cleaning rod is basically a long metal knitting needle. The rod is normally used to clean just the inside of the outer slide. That part of the trombone is polished inside, so it can't handle the abrasive snake. Take care, as rods are made of a moderately soft metal that is easy to bend.
I use cheesecloth or low-weave muslin to cover the rod. Cheesecloth is a very widely woven and porous cotton cloth, but anything similar will do. Bill Watrous (and most other people) keeps a plain white t-shirt in his case, and tears off a strip as needed. Pick 1.5 to 2 meters of something soft, disposable, and slightly absorbent.
When wrapping cloth around the rod, make sure to cover the pointy end, or it may scratch the inside of the slide. First, pass the cloth through the needle hole, then wrap it over the end once or twice. Then wind the remaining cloth down the length of the rod, covering the metal surface. Spare cloth can be loosely knotted at the end of the rod for future use.
With use, the cleaning rod cloth will gradually become dirty - just cut off the dirty bit and advance the cloth toward the needle hole.. If your cloth is coming out green or dark with every use, it is probably time for a bath!
Every time, use the cleaning rod and wipe off the inner slide before applying slide cream.