Before you can house the creatures you'll befriend you will need to build the Monster hutss. You can have up to 7 physical huts on your farmland. Each hut costs 1000 G and 100 pieces of Lumber. To order a hut talk to Camus in the village. The hut will instantly appear on the north side of your farm. The hut can hold 4 monsters.
Once you have the 7 huts built, you can continue to expand them to house more creatures. Each 1000G/100Lum will give a hut its basement. Each hut can have 3 basements, allowing you to house a total of 84 monsters.
If you choose to purchase Fodder, it can be found at Neumann and Camus' Farms for 100 G a piece. Unfortunately you can only buy 9 pieces at a time, so it will take a lot of purchase confirmations to get the amount you need. Once you have the Fodder you need to throw it into the bin opening on the outside of the hut. Each hut has its own holding bin just for feeding the monsters that reside inside.
Each of the beasts you raise on your farm will eat 1 piece of Fodder per day. There isn't any way to pasture your animals like in previous Harvest Moon games. Fodder seeds can be purchased from Jean's store for 600 G. It takes 10 days to mature and Fodder grass does not grow back. You will have to replant once you've harvested.
To capture monsters you'll first need to talk to Tabatha at Bianca's mansion. The maid will give you the Pet Glove tool, which allows you to pet your chosen monster until it decides to move into your huts. To use the tool, venture into a cave with monsters and press B while standing next to a monster. You don't have to stay in one place while you use the glove, so feel free to dodge the creature's attacks and then counter by petting it. If you're sucessfull, hearts will appear above the monster's head.
Once you finish taming your beast, you can choose a custom name for your new friend. After that it will immediatly teleport back to your farm and reside in the first available hut you have. There isn't a way to rearrange which hut your tamed monsters are living in.
Raise your monster's FP by brushing them every day. You can also harvest items if you have Livestock monsters, but only every 4 days. If any of your beasts are of the Working class then they will start their jobs at 10:00 am once you have asked them to help you with the farmwork.
If you take a monster with you into the caves, it will attack the nearest creature it comes across. The friendly beast won't attack Kikai machines, and if it runs out of HP it will teleport back to your farm. You won't loose any FP with the creature if it's defeated in battle.