Here;s a few ideas.
For rodents and small reptiles, bury a bucket up to the rim in the ground. Put a low fence or net or just two logs, making a "v" into the bucket. The critter follows the fence to the bucket, then falls in. Reptiles will simply fall in. Mice & other small rodents require some sort of bait -- sweetened grain works nicely. Once you've got one mouse in there, others will follow, however. They're attracted to the mice that are already trapped. This is the best method I've ever found for getting rid of gophers. Takes a big bucket, though, as gophers jump.
For bats, birds, other flying critters -- mist nets.
Variation on a mist net -- take a small fruit tree or a feeding station of some sort, and enclose it on three sides with a net. On the fourth side, prop the net up so the birds can come and go. Wait until there's a bunch of birds in there. Drop the fourth side of the net closed.
For fowl -- chickens, turkeys, pheasants, quail, guinea fowl, peacocks, etc. etc. etc. -- make a pen of netting. Put a trail of feed into the netted enclosure through the doorway. Get the birds used to eating in there. Then rush the doorway -- most poultry type birds are dumb enough that they'll flee from you and you'll have time to close the door. Note that guineas and quail are dumb enough to kill themselves flying into the netting, however.
For deer and other large mammals -- you can either A: get them used to coming into a pen for food, then shut the gate or B: chase them into a pen, using cloth panels or scary humans to direct them through the gate. "A" is probably simpler and less stressful for the animal -- large herbivores are, generally speaking, easily attracted to grain, salt licks, etc. In fact, my father has a cabin in a remote area & has elk come into his back yard simply for hay, despite a forest full of browse for them. They're comfortable enough in the presence of people that we can stand out on the deck watching them eat 10-20 feet away. Note that deer, elk, cow-like wild animals, etc. can all jump impressive heights.
For predators -- as previously mentioned, box traps. I like the kind with a trip of some kind attached to food, and a sliding door that drops down catching the animal inside. For most predators and scavangers, you'd do good baiting the trap with something smelly, like last week's fish heads. Certain animals -- weasels and mink come to mind -- are more successfully caught with live bait, like a small chicken tied inside the trap.
Generally speaking, predators are fairly easy to catch. There will be some too smart to trap, but there will also be plenty of dumb young ones who haven't learned better yet. (And I have caught coyotes in a plywood box trap when I was TRYING to catch a stray dog, so I KNOW this rule applies to 'yotes.)
The FUN part is getting the predator OUT of the trap.
Leva