I am going to go under the assumption you will first be doing schooling, unrated shows at first.
For the Hunters, a navy coat, white show shirt and tan breeches is a classic look that will never go out of style. You don't see black coats in the hunter ring, it is more of a jumper thing. Tall boots, black gloves, and a black (preferably velvet type) helmet are also going to be necessary. No boots or polos are allowed on your horse in the hunters, and a standing martingale is ok for jumping but not for flatting. Use a shaped, white hunter pad. As far as bits go, most people show in some type of snaffle (dee, full cheek, loose ring), some people ride in kimberwicks or pellhems. Gags, elevators, etc are not allowed in the hunter ring as well as figure 8 bridles and flash nose bands. For the hunters you want to make sure both you and your horse are clean and tidy looking.
For equitation, pretty much all the same for the hunter ring applies but you are allowed to use boots on your horse with the exception of bell boots.
The jumpers is different. Most types of boots are allowed on your horse, running martingales are allowed, standing martingales are allowed up to a certain height (I think its up to 3 foot but I haven't checked the rule book in a while), you can use a square saddle pad of any color and are welcome to show in a polo in a less formal setting. A wider variety of bits are also allowed in the jumper ring.
This is kind of a quick over view and are according to US rules but I am assuming it is similar everywhere else. Just make sure to check before hand.