This Guide will be broken down by Animal Name. Then Food type and finally duration it take to tame the animal. And how to acquire the Food for the animal.
As always tamed animals need a home. So make sure you have crafted the proper house for your new critters before you bring them home.

Dog ------ Pet them each day to gain maturity levels.----- 5... "Who's a good boy?!"
Chicken ------ Bird Feed ------- 7.... Craft with Wheat or Buy from Mr. Jones
Bunny ------ Carrot ------- 7..... Grown or Harvested from fields.
Sheep ------ Hay ------- 10....Crafted at Barn drying table using Tall Grass.
Duck ------ Bird Feed ------- 7......Craft with Wheat or Buy from Mr. Jones
Goat ------- Hay ------- 10...Crafted at Barn drying table using Tall Grass.
Cow ------- Hay ------- 14...Crafted at Barn drying table using Tall Grass.
Horse ------- Hay ------- 14...Crafted at Barn drying table using Tall Grass.
Cat ------- Fish ------- 5..Catch them Fishing. Silver in color.
Pig ------- Potato ------- 8..Grown or Harvested in Fall.
Dung Beetle -- Beetle Feed ---- 8.. Craft Food from Poop.
Fox ------- Small Fish ----- 10.. Catch them Fishing.
Bear ------- Honey ------ 15..Collected from Bee Hives or Apiary.
Hippo ------- Hay ------ 15..Crafted at Barn drying table using Tall Grass.
Deer ------- Tall Grass ------ 14..Grows Wild in Forrest Area.
Ferret ------- Wool ------ ?...Collected from Sheep.
Penguine --- Small Fish ------ ?...Catch them Fishing. Red in color.

Hope it helps!

Found a great resource for the animals on the Garden paws Wiki page. I have
linked it below. Have fun!

https://gardenpaws.fandom.com/wiki/Animals

Another great wiki page for Taming and raising your animals!!

https://gardenpaws.fandom.com/wiki/Taming_and_Raising_Animals