Answer:
English riders often wear jodpurs or breeches and English boots and a helmet or hard hat. They may also carry a whip/crop/riding bat and/or English spurs, which have no rowels as western spurs do.
Western riders may or may not wear a helmet. Beginners often do, while more advanced riders may not, although i advise a helmet for riders of all levels, unless you are in a show which requires you don't. They also wear jeans, maybe a cap or cowboy hat if they have no helmet, and they ought to have cowboy boots with a smooth sole so you could easily slip out of them should you fall off instead of getting hung up in the stirrup and dragged.