Well, most obviously, you clean it after every ride – just wipe it down with a cloth. Something you can do with your bit is boil it! It not only sanitizes it, but it also makes it sparkly clean; Just remember that the bit is very hot when it comes out of the pot, so take it out with tongs, etc.
I've heard of/seen a lot of bit "polishes," but I'd have a hard time with putting chemicals like that into my horse's mouth.
The hills were bathed in moonlight, the shadows not so stark;
Silver light reflected off his brown hide as he held me in the dark
– I love you, Mudpie!