OK this is what you would have in a perfect world lol.
Horse
Brown cavesson bridle with flat hunter noseband and pelham or double. The browband will be either flat and plain to match the noseband or a velvet browband if your horse is the right type for hacks/ riding horse classes.
Brown show/ working hunter saddle with brown leather girth and either no numnah or a discrete one which fits the saddle perfectly.
Rider
Tweed jacket (brown or green are most popular) shirt (not white pale blue, cream etc) tie with tie pin, hairnet and hair in bun with a navy velvet hat with flesh coloured straps, cream/beige jodphurs, long black leather boots with garter straps, brown leather gloves and cane.
Something like this (plain browband if horse is a hunter type).
ETA. Some prefer an in-hand bridle. This photo shows how the handler is wearing pale trousers because the horse has dark legs. If your horse has big white stocking, you might want to wear dark trousers.