>>414253First, check your local regulations. You'll typically have to obtain some kind of certificate or license. In Kansas you take a test online to receive a certificate that will allow you to purchase a license. It's only like 20 and some change.
As far as traps go, some number 3 coil spring footholds will probably do you just fine for both coyotes and foxes. You can try live box sets, but I've never had any luck with trapping coyotes. Also, I've never had luck trapping foxes period, although they aren't exactly abundant where I'm at.
It's hard work too, especially if you are going to be running multiple sets. A lot of driving stakes and hiking in cold temperatures. Depending on the type of soil and temperature, you can expect to spend a large amount of time trying to find a good spot to drive your stakes without hitting rocks too big to drive through. Be sure to bring something to snack on and a drink or two.
Try to find some one to tag along, especially if you are going to be in an extremely remote area. I trap on my dad's land, which is about ten miles from the nearest town. I usually have some one come with me just as a precaution. You don't want to be strand out in the middle of nowhere with a broken leg, or stuck in the middle of a creek because you got your hand stuck in a 330 conibear water set. Conibears are body traps btw, illegal in some states, but not in Kansas. They are powerful traps, and usually kill the animal instantly by crushing the throat or the chest. I've only ever used them on coons and beavers. A water set, as mentioned earlier, is any set that you put in or under water. You can use foothold traps and certain drag systems that will drown the animal.
Don't forget to dip or dye your traps. This prevents corrosion. There are a number of other measures you should take when setting your traps, far too many to mention here. I'd suggest finding some online trapping forums. They'll get you started off on the right foot.