Definitely Harry Potter! My kids will be as familiar with The Tales of Beedle The Bard as with Cinderella and Snow White. :D And Hogwarts will be another home for them, as it was for me. They'll also watch the movies over and over with me. <3
I'd also read or recommend to them the Narnia books, and The Hobbit (and we shall watch the Lord of the Rings movies together too; I'll let them read LotR when they think they're old enough). I have loved these books since my own childhood and they're definitely classics for children. I also loved Pride and Prejudice, Jane Eyre, Great Expectations and Oliver Twist from the age of seven or eight, and I'd start them off on abridged versions and then let them read the unabridged ones when they want to. Also, I grew up reading loads of Enid Blyton. I think the Famous Five series was my favourite; I had the entire set. My kids will grow up on those too, and The Secret Seven, The Five Find-Outers and Dog, Malory Towers, St. Clare's, and her Rewards short story collections.
And, of course, mythology. I grew up reading loads of Norse, Greek and Egyptian myths, as well as Hindu myths. Obviously, as a child I only had access to the censored versions; therefore I believed that the part of Osiris's body that Isis could not find was his leg, only finding out years later that it was his, ahem, phallus. XD
Some gorgeous Potter-related graphics that sum up my feelings perfectly:
^ There are a couple of typos, but I still think it's gorgeous.
^ I thought that was the most beautiful scene in the Half-Blood Prince movie: when all the students and teachers of Hogwarts, standing by Dumbledore's body, raise their wands together, and the light from all their wands dispels the Dark Mark in the sky.
And that is the sort of thing I want my children, nieces, nephews and godchildren to read. Books that teach them to believe in magic and the impossible. Books that teach them the importance of friendship and family and bravery and loyalty and faith and love and forgiveness and ideals, that provide strength and show them that you can use your brains to get yourself out of sticky situations (I <3 Odysseus, even if he is a bit of a dickhead at times). Stuff that instils in them a sense of awe and love for the ancient civilisations. Stories that develop a love for adventure and nature.
One of the reasons I am so fond of Potter is all the references Rowling makes to creatures and legends from all over the world; it led me on a merry treasure hunt to find all these things, like manticores and divination methods and alchemy and etymological roots and so on, and I learned so much. (It's also one of the reasons I <3 The Dresden Files - the way Jim Butcher makes use of so much fantasy lore - but it's hardly for children.) The series isn't perfect, of course, but if you asked me what I would read to my children, it's the first thing that comes to mind.