Coyote Hunting takes some practice,as Firefighter mentioned it would help in getting some gear for Predator Hunting in general.Some items You may already have just from Hunting other Game Animals such as Deer,Turkeys,Antelope Elk etc..!

Shotgun with buck shot or a smaller caliber Rifle if your going to keep the pelt,have it mounted or give it away to a Trapper?
Camo Clothes
Predator Calls (misc mouth calls)
Electronic Callers (this can get as expensive as You want to go)
Scent-free washing detergents and Scent-killer sprays
Misc Decoys,like the foam Rabbits,full-size Flambeau Coyote decoy,Mojo decoys that move,wiggle or shake,HS Feather on a foldable stick for a visual attractor
Cover Scents,Fox Urine,Coyote Urine,Skunk Urine etc....!

You can go as far as You want to Hunt Coyotes/Predators or as cheap as possible,the main thing to remember is a Coyote uses its keen eye sight and sense of smell to detect what they are hunting and what is out there hunting them so You need to sit still,hunt the wind and stay as scent-free as possible when hunting them.