Because of Winn-Dixie by Kate DiCamillo - Winn-Dixie is to Opal as Sandy is to Little Orphan Annie.
The Henry Huggins series by Beverly Cleary - Ribsy is Henry's fun-loving dog. He once gets lost, but then he is found, and now I'm unintentionally singing Amazing Grace.
Upper Elementary School
Coraline by Neil Gaiman - The cat rules. He would love Emily Strange's cats.
May Bird and the Ever After by Jodi Lynn Anderson - Somber Kitty is precious.
The Incredible Journey by Sheila Burnford - Two dogs and a cat who travel quite a distance and overcome many obstacles to find their owners.
Middle School and High School
In the following book series, dogs are able to communicate telepathically with their owners. I don't want to spoil anything about either series, so I'll just say that they are quite different series with quite different doggies, and The Fallen = angels while Wizards = magic.
The Fallen quartet by Thomas E. Sniegoski
The Young Wizards series by Diane Duane
For a total change of pace, try these comedies with canine capers:
Boys are Dogs by Leslie Margolis - A middle school girl uses what she learns from her dog training manual on her classmates as well as her puppy. This is the first in a series of books about Annabelle and company. I recommend the entire line. For ages 11 and up.
Leslie Margolis also writes the Maggie Brooklyn Mysteries, in which the title character is a dogwalker who accidentally becomes an amateur detective. This is also a fun series. For ages 11 and up.
Stray by Stacey Goldblatt - The daughter of a vet and a behavorist wants to live by her own rules. For ages 14 and up.
If you and your little ones want to avoid danger and worry altogether, try these funny stories which feature pets as supporting characters. All of the following are appropriate for elementary schoolers.
Judy Moody by Megan McDonald - A cat (named Mouse), a toad, and other critters. Judy's brother, Stink, now has his own series, which is aimed for slightly younger readers.
Henry Huggins by Beverly Cleary - Henry has a dog named Ribsy, who even gets his own book!)
Socks by Beverly Cleary - A cat who doesn't understand why there's a baby in the house.
Harry books by Gene Zion and Margaret Bloy Graham - Cute dog, cute stories for beginning readers.
Various American Girls books - Most of the girls in the American Girls historical fiction series have pets. For example, Felicity has a horse and Josefina has a goat!
The Baby-Sitters Club by Ann M. Martin - Mary Anne has a kitten named Tigger, and plenty of the other baby-sitters and baby-sitting charges have pets.
The Penderwicks by Jeanne Birdsall - Hound is a loyal dog and a valued member of the family in this book and in the sequel, The Penderwicks on Gardam Street.
Yes, there are plenty of books that do not offer happy endings for the pets - some which I think are well-written and some that I think are highly overrated - but I'm not talking about those right now. (Sorry, Justine.)
Note that His Dark Materials by Philip Pullman, which is for an older audience, defies category due to its inclusion of daemons, shapeshifting animals that are a mixture of person's pet, best friend, soul, and twin. (Kind of like Twinkie, though she can't shapeshift.)
My favorite literary critter companions of all time, curiously, are not domestic:
Pantalaimon the daemon from His Dark Materials trilogy by Philip Pullman
Falkor the luck dragon from The NeverEnding Story by Michael Ende
Fiddlestick the dragon from Strangewood by Christopher Golden
Dot the laptop from The Young Wizards series by Diane Duane
Related Booklist: Cats Cats Cats