I loved “The Book of Night With Moon” by Diane Duane. It’s about cats, written from a cat perspective. Oh yeah, the cats are all wizards, too. 
Lassie Come Home. Really.
I have a vague memory of reading Big Red or wanting to read it when I was younger and not getting around to it.
I’ll add to the list for anyone else who may be surfing the thread looking for animal books to read.
Nonfiction: The Good, The Bad and the Furry a guide on the highlights and lowlights from the most popular breeds. This is a great reference that is funny and readable.
Dogs Who Found Me. An upcoming dog writer, IMHO. I haven’t read his other stuff as of yet.
There is an out of print book that I devoured as a kid that was called something like, The Dogs that came to stay. Set in NH or VT ( maybe CT.) and a man and his wife take in these two stays ( spaniels/retreivers, IIRC) and how they started out living in the wood shed. There is an event with one of the dogs digging up shot possums ( or skunk) that had been buried and rolling in that mess that has stuck with me for years. Can’t remember the author. He was a writer by trade, but not overly known.
This is a new release and I am waiting for it to come into the library: Dewey. A cat who lives in a library. What is not to love already?
Kid’s Book: Six Dinner Sid Picture book.
Jenny and the Cat Club and other cat books by Ester Averill. Delightful cat stories.
Golden Cat Haven’t read it. Mean to one of these days.
Well, duh. Everyone knows that!
These are all awesome and I have put them in my library book queue.
This book came in first from my large request from the library interloan and I am thoroughly enjoying it. Not overly sappy, but Camuti isn’t a sappy kinda guy.
This is a bit off the cat and dog track and into exotics, but nobody should miss My Family and Other Animals by Gerald Durrell. He wrote other books about his love of animals and the zoo he eventually came to run, but this was the best.
Alex and Me
http://ebookstore.sony.com/ebook/_/R-400000000000000096960
I, too, have “Dewey” in my library queue!
“Misty of Chincoteague”
“Sounder”
“Old Yeller”
“Where the Red Fern Grows”
Enslaved By Ducks is a wonderful little vignette into a writer’s life and his life with a variety of animals, including ducks and rabbits.
Some funny observations and set in Michigan YAY for MI!
These aren’t entirely dog/cat books (as a country vet, Dr. Herriott treated plenty of horses and livestock too), but I love the “All Creatures Great and Small” series.
Another vote for Nop’s Trials- by far the best dog book I’ve read, and it’s for grownups. It has a sequel, Nop’s Hope that is also very good. I liked it even before I became a working border collie fanatic.
Seconded.
I’m embarassed to ask this, but could somebody tell me which of the above books are NOT SAD? Because, you know those animal books. I’ll never forgive Sounder for effectively killing the dog twice.
Zsofia - my suggestions:
The Dog Who Wouldn’t Be: Bittersweet, or sad if you prefer.
Big Red: Happy, sequelae are of progeny
White Fang: Happy
All the Kjelgaard “Red” books have upbeat endings. The Terhune ones are usually upbeat and are at worst bittersweet. The James Herriot stories are all over the map, but only rarely depressing.
ETA: Oh, yeah on preview OtakuLoki’s suggestion of White Fang is also a good one, though it wades through a couple of pretty tough acts to get there.
My absolute favourite book as a child… it still brings a tear to my eye… it is definitely bittersweet though.