I use, and recommend, about 20 gauge wire for snares. You can use it for small ground squirrels up to coyote-sized critters. For small stuff like squirrels, the wire attached to a carved wooden stake is enough. For larger game, tie the snare wire to 550 paracord, then tie the paracord to the wooden stake. These are simple snares that do not use an "engine" to work (an engine being a twitch-up or other trigger method that uses the energy of a branch or other kinetic means). A simple snare only uses the critter's momentum and direction to work.
As long as you know and recognize sign and tracks, you can easily snare critters. Think about their head size, and the height at which they normally plod along. Then you can set the loop to be just big enough for the game you want, and you can set it to be at the height their head will naturally go through when they are "walking."
Things to consider: de-scenting via smoke (critters are not alarmed by the smell of campfire smoke in many places as they are used to campers), or rubbing mint leaves on yourself and the wires/wood/paracord; also be prepared also that your snare may not kill an animal, only harm it irreversibly, so you may have to finish it yourself. If you have not been in the position of having to kill an animal with your knife or other up-close method, it can be very disturbing. The first time I had to behead a squirrel I snared, I held it up by the tail. It twitched its legs and arms (the head was removed) and kept up these convulsions for what seemed like an eternity. So, be ready.
Constructing a snare is easy. It's placement that is the big challenge.
