Choose one type of animal at a time like "Horses" or "Cats" or "Dogs." Read online tutorials on drawing that animal. Take some photo references of the animal and also do some quick sketches of it. If you look at a live animal, say a sleeping cat, and try to draw it in under three minutes -- and then do that every time you see the cat sleeping -- eventually your eyes, your mind and your hand will get used to all its poses and proportions.
I've been studying my cat by taking reference photos and drawing him, usually while asleep but sometimes when he's sitting up awake, for several years. The more often I draw him, the more realistic he comes out. Realism takes practice and observation.
Good books can also help. I recommend "How to Draw Animals" by Jack Hamm, it costs $10 or so from Amazon or assorted bookshops and it's a Dover book. Don't even try reading through it except to skim and get an idea where things are. Work through it by section one animal at a time copying Jack Hamm's drawings until your copies look good like his. Then try drawing the real animal.
Practice, observation and study are the ways to draw real looking animals. Books like Hamm's are great for learning how to do textures for fur and feathers and scales like on a bird's feet or a snake. Getting eyes right means studying them for that type of animal. Put a cat's eyes on a horse and you have a weird monster (heck, give it fangs too and call it a good monster!) Same if you give it human eyes, it will be scary.
Cats have very short legs compared to people or horses. I began studying horses last year when I started getting comfortable with cats and they are so different. The joints in any animal's legs are different from the ones in human legs and that takes practice to know which way they bend and where.
One quick answer is to take a good photo of one, trace the outlines, then draw from the photo very carefully shading dark and light wherever it is even if you think it looks weird, wrong or stupid. It will come out right if you do it carefully. Try printing out the picture and marking a grid over it, then put a grid on your drawing paper and draw only what's in each square of the grid in order where that square is. This gives even total beginners great accuracy and real looking animals -- and it is good practice for getting them right without the grid too later on.