Merchant missions.
You should always be able to get exactly the right amount of coops, cages, baskets and boxes that you need to fulfill your order from any merchant. It is usually advisable to get a single merchant voyage, propose and accept that mission then "draw stores" from the merchant that sold you the voyage.
Now plan your voyage using either knowledge you've gained from previous animal seeking quests or a "cheat sheet" like this one - SoT map. I tend to find it best to go to the bigger islands as there are usually more animals.
To actually catch the animal pick up the correct receptacle (coop for chicken, cage for pig or basket for snake) and go ashore. Once you spot what you're after chase it and when you're close enough for the writing to pop up e.g."Golden Chicken" press and hold RT or left mouse button for a few seconds until the chicken or pig is caged. Now return to your ship, not forgetting to feed you pig when he cries.
Snakes are a bit trickier because they bite but at least they don't run. Approach them slowly, whip out your instrument (a musical one) and play them a quick tune to send then to sleep. Now put them in a basket and return to the ship. Keep an eye on them and if they start to wake up drop them and play another tune.