What are three ways you should handle animals

Depends on whether they are tame or not.

Domesticated But Not Yours: Curl your fingers (either make a fist or just a half closed fist) and let the animal sniff you to get them familiar with you. This works for Dogs, Cats, Horses etc. After letting them sniff you, gently stroke them. Don't just jump right in and hammer them with pets. If they give you a warning (growl, hiss), this includes dogs and cats, then back away. Let them have control over where you touch them, and how you touch them. It's all about trust.

Undomesticated, Wild, Untame: I do not recommend approaching wild animals, they are dangerous and can/will hurt you. But if you must approach them, do it with extreme caution. Wild animals are unpredictable and dangerous. Again, if you get close enough, let them sniff you. Do this with caution they may bite. Then, if they let you (if they growl/hiss BACK AWAY) very, gentle and calmly stroke them in slow motions. You do not want to startle them. If you wish to take the animal home with you (because it's hurt, starving, cute) bring out some food. Again, approach them with caution. And hold out the food for them to take, generally (from my experience) the animal won't approach you, or at least won't get to close.