I would reccommend you started with the racoons and fox before trying to graduate to coyotes. Coyotes are hard to trap and you will get frustrated if you have no trapping experiece. I like Sleepy Creek 1 1/2 OS modified coil springs. These are a great little trap and work very well for coon and fox. If you are going to try the coyotes I recommend MB-650 IL, yotes can' t tear them up. They are a little hard on the pocket book, but well worth it.
You will need a digging hammer, a heavy duty trowell, a dirt sifter, a whisk broom, pan covers or pads, urine bait and lure. As for stakes i prefer the berkshire disposable stakes, otherwise you can make your own out of rebar with a nut and washer welded on.
I would highly recommend you get the books The Dirthole set, and The Flat set by Charles Dobbins. These are invaluable to the beginner. He goes through equipment, as well as how to set and where to set. Farmland Fur Trapping by Bernie Beringer is good as well for a general book. That one will help you with the coons.
Hope this helps. The best advice I can give you is to get those three books and go from there. Good luck and let me know if you need anything else.