This video starts out discussing some common issues that are caused by not keeping a trumpet clean and lubricated, and then shows how to clean your horn.
If you want cleaning brushes like the ones Monette uses, it's these:
Valve casing brush - The picture is wrong - it shows a mouthpiece brush. But this is the one used in the video, and it is your best friend for cleaning your valve casings. You can use it on lots of the other tubes on your horn as well, even though it looks too big. The bristles are soft enough that it fits into smaller tubes and does a great job.
Mouthpiece brush - Useful for things other than mouthpieces. Works great on piston ports, and you should brush out the mouthpiece receiver on your horn with it as often as possible because that's where lots of gunk tends to accumulate and cause problems with mouthpiece sealing and mouthpiece shank wear.
Snake - It says it's for trombone, but trust me - this is the one you want for trumpet as well.