In my experience, the best way to teach animals and their sounds to kindergarten and preschool children is through songs, flashcards, playing, and drawing.
My favourite website is Super Simple Learning because they have a wide variety of free resources (videos, flashcards, colouring pages, worksheets, and crafts). In particular, the kids like very much and Old McDonald had a farm and I have a pet:
I’ve found these flashcards here: http://supersimplelearning.com/resource-center/browse/song/m-o/old-mcdonald/
One song the kids can’t stop singing is 5 Little Monkeys Jumping on the bed (Little Fox version):
Super Simple Learning has it too with a lot of free resources, I printed and coloured this play set:
Two weeks ago they coloured these:
Source: ISL Collective
And last week I printed these out:
… And they coloured them :-).
Then, they learn a lot playing BINGO and with little flashcards.