Writing about horse equipment?

i’m writing a story about life on a farm and i am sadly lacking in horse lingo. when you’re getting ready to ride a horse, i know you’re saddling up (or something like that) but i don’t know much besides that except that:
a saddle is the thing you sit on with the horn (which you hold) and the stirrups which you put your feet in.
a bridle is ???
reins attach to the bit (goes in the horse’s mouth) and are used to direct a horse unless it’s trained to respond to your movements. (what’s that called?)
a lead rope is a rope you put on the horse (around the neck, or connected to the bit?) to lead it around (always on the horse’s left)
what else is there?
thanks.
thanks to the first answerer, but you added some more questions.
headstall?
halter? i’ve heard that one but i forgot to mention it.

and it’s western riding. thanks for the help so far

saddling up (tacking up)
horn (saddle horn)
bridle (attached to the bit & to the headstall)
responding to movements (trained to use feet)
lead rope (attached to halter to lead horse around with)
you might add more details about if you’re riding english or western because there’s a whole different set of tack for each saddle.
hope i helped