The Phantom Tollbooth is a must, should you wish your children to grow up being cool.
The Moomin stories by Tove Jansson. We all have our favorites, so I’ll just say get ALL of them.
And Lewis Carrol’s stuff too, but you already knew that. 
I grew up reading the Little Golden Books. I think the Roly Polly Puppy was my favorite too. I also read a lot of Bearnstein Bears books and Little Critter books.
Also, The Giving Tree and other Shel Silverstein books. And my favorite book as a kid, The Haunting of Grade Three by Grace Maccarone.
Gerald Durrell’s autobiographical stuff - most especially My Family and Other Animals.
Roald Dahl
Tin-Tin by Hergé (skip …in the Congo) - Nice complement to Asterix when older.
The Color Kittens was one I remembered from when I was a toddler and managed to find and read to my girl.
All the Pooh Bear books - preferably the AA Milne version.
Wind in the Willows is great for reading out loud - I have the copy my mum read to me and I read it to my kid when she was about five years old. She wanted to be Toad when she grew up! (I wanted to be Ratty, but suspected I’d end up as Mole.)
Dr Seuss too, for instilling a love of language.
I’ll second the Gerald Durrell books, Herge & Lewis Caroll.
Also, see if you can get a well illustrated copy of the *unedited *Peter Pan, there’s this great little sentence in it about pink fairies being girls, blue fairies being boys and the little purple fairies that aren’t quite either. I’ve only seen that version once, but what a great way of introducing the idea of diversity to kids.
The Tom Swift Jr series of boy’s novels.
http://www.tomswift.info/homepage/
Incredibly cheesy, but very fun.
Mike Mulligan and His Steam Shovel and Katey and the Big Snow by Virginia Lee Burton
My own favorite Golden Book was The Saggy Baggy Elephant
My son loved (and still loves at the venerable age of 10) the original Thomas the Tank Engine stories by Rev. Awdry.
When he was very very young he also had some beautifully illustrated books by Alexandra Day about a rottweiler named Carl; Good Dog, Carl, Carl Goes to Daycare, and others.
Are people not mentioning Dr. Seuss because it’s just assumed you have them? I’ll put in a vote for my own favorite, Fox in Socks
Some of my very favorite childhood reading came from My Bookhouse Books The set I have had been my grandmother’s in the 20’s. Volume one is Nursery Rhymes and Volume 12 is called Halls of Fame, with works by Chaucer and Tennysin and other “heavy hiitters”. Finding a set of all 12 together in good condition from what I understand isn’t so common, but I’ve seen at least a few volumes in lots of used book stores. They really are treasures, with absolutely wonderful illustration.
This book is in the realm of Shadow Castle and The Westing Game.
The Diamond in the Window
I read it and reread it.
But E. Nesbit was my absolute favorite. She created worlds that I wanted to inhabit.
This piece by Gore Vidal discusses her work in particular, and writing for children in general.
The Writing of E. Nesbit
Edward Eager was sort of a disciple of E Nesbit. It was from reading his work that I discovered Nesbit.
Edward Eager
Isaac Asimov had a young reader series called Lucky Starr, Space Ranger which was a great intro to the world of SciFi. I even see elements from these Asimov books in the Clive Cussler (?sp) novels.
I loved these books as a kid - there were 5 or 6 in the series.
I was also going to recommend Mike Mulligan and his Steam Shovel but I was trumped. This is the only book I can remember my Dad reading to me. (I know he read alot to me but this is the one I remembered).
Not quite what I wanted, but HongKong, ya gotta raise yer own kids, and mere Dopers can’t do everything for you. 
There are even movies made from this stuff.
There was some mention of kids making up their own stories. This Allsburg guy made a book wherein, the righthand page was a suggestive picture, the lefthand page was blank, so the lil’ dears could get busy. For some reason, I can’t recall the title. Never used it myself, but I DO recall pushing it when I worked at a bookstore.
It’s hard to find, and somewhat expensive if you get it off Ebay, but Munro Leaf’s* Gordon the Goat is a perfect book. The illustrations are simple line drawings, but the words are what’s important, and it gives a wonderful message: Love who you are, and don’t feel you always have to follow the crowd.
*Munro Leaf is also the author of Ferdinand the Bull but Gordon is much shorter and easier for a child to grasp.
*Frog And Toad Together
The Giving Tree
Jacob-Two-Two Meets The Hooded Fang*
The very first real book I ever read was in 4th grade, that wasn’t a choose your own adventure, was How to Eat Fried Worms. After seeing it on the libraries list of “Banned or threatened” books I read it again a few years back.
Marc
*The Prydain Chronicles *by Lloyd Alexander:
The Book of Three
The Black Cauldron
The Castle of Llyr
Taran Wanderer
The High King
I read *The Book of Three *in sixth grade, and gradually (lazily) worked my way through the rest over the years, finishing *The High King *early in my freshman year in college. *Taran Wanderer *is my favorite; it’s the classic “hero coming of age” theme, only done in an unusually elegant way.
They’re definitely children’s books, but I’ve always loved Alexander’s outlook on life, death, honor, happiness, maturity, and hope. Do wait until your kids are a bit older (I’d say 10-12 years is a good age to start). Alexander also wrote The Foundling, which was sort of a prequel book to the series. I suggest saving that one for last, however.
Taash and the Jesters, by Ellen Kindt McKenzie.
I don’t know how well know it is but I read it in either later grade school or early junior high, and just loved it. Magic, danger, adventure, a lost baby prince that and orphaned boy helps take care of. If your child is of an age to enjoy Prydain they’d like this book, it has a similar feel.
Seconding Mike Mulligan and his Steam Shovel.
Caps for Sale is another of my favorites.
The books by Beverly Cleary, especially Ramona and Beezus, are good if you have girls. They capture perfectly the trials and tribulations of sisters.
I forgot about this book, and I in fact have a copy (hardcover to boot!) with me even though I have no children.
It was my absolute favorite book, and it always made me cry.
One of my most treasured books when I was a little girl was Silver Pennies, a collection of poetry edited by Blanche Jennings Thompson. Unlike many children’s poetry books, this one includes a lot of dark, thought-provoking poems along with the whimsy and fluff. Every one of these poems is a gem. I still read my copy of Silver Pennies.
Another vote for the books by Enid Blyton. I read and enjoyed the Adventure series which featured four English children; Jack, Dinah, Lucy and Philip and their parrot, Kiki. Written in the thirties they are quite dated now and a bit politically incorrect in places, but still enjoyable.
A more adult book that children can’t help enjoying is Robert Louis Stevenson’s Treasure Island. It’s hard to realize what a controversial book this was when it was first published. It had a cheerfully amoral villain (Long John Silver) for the hero, who escaped his deserved punishment at the end.