Another vote for the “Anne of Green Gables” books.
But I would forget “Little House” in favor of the much more vigorous Caddie Woodlawn.
The Betsy-Tacy-Tib stories are a must.
Yay, I get to mention my favorites!
For the nature-lover, you can’t beat Jean Craighead George’s books. Julie of the Wolves is probably her most famous (and won a Newberry, IIRC), but many of her books revolve around strong female protagonists and their relationships with the natural world. (The remainder of her books revolve around strong male protagonists and their relationships with the natural world).
For the fantasy reader, you gotta check out The Hero and the Crown, by Robin McKinley. While all her books are fairly good, and all (IIRC) have female protagonists, not all of them are appropriate for children. The Hero and the Crown is a classic sword-and-sorcery novel with a buttkicking heroine; it also won a Newberry.
Those are my fvaorites. Susan Cooper’s The Dark is Rising series has a bunch of leads, some of whom are female, but I think the main protagonists are male. Le Guin’s Earthsea series alternates between male and female protagonists until the fifth book in the series, when it switches to an ensemble cast. And for older kids, Phillip Pullman’s His Dark Materials trilogy has a great female lead.
Someone recommended Jacob Have I Loved. I’ll extend the recommendation to most of Katherine Paterson’s works: they tend to have female leads and are very well-written.
Finally, you could do a lot worse than Mildred Taylor’s Roll of Thunder, Hear My Cry. It’s a harrowing account of a Southern Black farm family in the early twentieth century, beautifully written and heartbreaking.
Daniel
Oh, I almost forgot, the American Girls series. I’d say that is for 8, 9, & 10 year olds. The series follows the lives of maybe 15 different girls living in America at different times in history.
Harriet the Spy! I loved it when I was a kid. Harriet is a wonderful sort of anti-heroine–she’s not pretty, she can be mean, and, of course, she spies on all her neighbors and classmates, but she’s really brainy and brave and fun.
Joan Aiken also has a wonderful series of adventures set in 19th century New England starring Dido Twite, a very clever and resourceful girl. Nightbirds in Nantucket is the one I remember best–it involves an obsessed captain searching for a giant pink whale. 
I have to agree with what most others have recommended. The Betsy-Tacy-Tib books are especially good. My dad used to read them to me when I was a wee lass, and I still love them.
There’s a great list of them at Heartless Bitches International (don’t worry, the site isn’t as harsh as the name suggests). http://www.heartless-bitches.com/culture/kidsbooks.shtml There’s lists of adult books with strong female leads as well as movies and women singers, too.
I read this for a children’s lit class I took years ago and loved it so much I had to buy a copy for myself. The title is Don’t Bet on the Prince: Contemporary Feminist Fairy Tales in North America and England by Jack David Zipes. They’re all fairy tales with strong female characters and male characters who recognize their strength and support them as equals.
I just wanted to come in and say “yay! Someone else loved “Ballet Shoes” by Streatfield!” I never owned it, but my school library always had it available (foolish students didn’t know what they were missing), and I took it out several times a school year. I still remember the light yellow hardcover, with the red lines on the spine spelling out the title and a wispy pair of pointe slippers, and how each line was slowly fading and being rubbed away each time I took it out, but you could still make out what it said if you looked hard enough…
The Anastasia Krupnik books by Lois Lowry are good.
Also, while Harry Potter might be the star of his epynomous book series, Hermione Granger is easily as strong and smart as Harry is. Definitely a book series to read if you want a strong female heroine.
The True Confessions of Charlotte Doyle by Avi
Who Stole the Wizard of Oz? by Avi.
His best books for other audiences are Sometimes I Think I Hear My Name and No More Magic.
Seconding recommendations for The Witch of Blackbird Pond, Island of the Blue Dolphins, The Dark is Rising series and the Anne series.
Also, Streams to the River, River to the Sea by Scott O’Dell (Island of the Blue Dolphins) is the story of Sacagawea, the Native American woman who acted as a guide for Lewis & Clark.
For the offbeat, modern fairytale, try Emma Donoghue’s Kissing the Witch and Francesca Lia Block’s books (Weetzie Bat, Girl Goddess #9, etc.).
The Young Wizards series by Diane Duane ROCKS–strong female characters, intense storyline. The first one in the series is So You Want to Be a Wizard. This is the most awesome set of books–highly underappreciated IMHO.
For more, check out Great Books for Girls by Kathleen Odean.
Most of my favorites have already been mentioned, but my all time favorite hasn’t: it’s The Upstairs Room by Joanna Reiss. It’s about a young (maybe 10 IIRC) girl in Holland during the Nazi occupation who goes into hiding with a Dutch family. I enjoyed this book more than The Diary of Anne Frank (perhaps because this girl survives, which was important when I was 10 and first read them both).
It’s very well written in a sort of choppy style that works well for a younger reader. The characters are real–they’re not absolute heros but have their own faults and weaknesses. However, they all show the power of the human spirit and the strength people can muster when forced to by circumstances.
Very funny, Dinsdale…I guess sitting at home in Rivendell, sewing jewels on banners with your mouth shut makes for a dull movie.
Now that I think of it, does she actually get even a single line of dialogue in the books proper? I know she speaks in the Appendix–maybe two lines when Elessar is dying–but I don’t think she gets a word out otherwise.
I recommend checking out Jennifer Roberson’s fantasy series about Tiger and Del, SwordSinger through SwordSworn, for older teens. [Isn’t it sad how many fantasy/scifi books we’ve had to recommend?]
Also, as previously noted, anything by Robin McKinley but definitely read them yourself first for age-suitability. She does not aim her writing for youth though her publisher keeps slotting her books into those sections at the bookstore, and you probably don’t want your ten-year-old reading Deerskin.
I popped in to mention Tamora Pierce’s books. Others include the Immortals series and the Protector of the Small series. For strong female characters, I’d go with these over the Alanna series - the second one I mentioned in particular. While the Alanna books do have a strong female character, she hides her identity as female through the first two books. The Protector books cover the next woman to become a knight in Tortall - with everyone knowing she’s female from the beginning.
I also like Gail Carson Levine’s Ella Enchanted and The two princesses of bamarre (this may just be titled The Princesses of Bamarre, I’m not sure). Both have great female characters.
I obviously misunderstood the “3-late teen” designation. You meant “age 3 to age 17 or 18 or so”. Expands things enormously. I don’t think anyone’s mentioned Ramona and Katie John (Beverly Cleary and Mary Calhoun, respectively).
More sf: Tanith Lee’s Black Unicorn and it’s Gold and Red sequels.
Another by Susan Cooper (author of the Dark is rising series) is Westward. The story balances nicely between equally strong male and female leads, who wind up working together for the same goal. Intensely imaginative book.
I’ll also vote for L. M. Montgomery’s Anne of Green Gables and Emily of New Moon series. Actually, any book by that author features strong female characters, and are all the more refreshing for having been written and published when Women’s Lib was just an idea.
For the older girls, Marion Zimmer Bradley has some great female characters, particularly in The Mists of Avalon*. Also for older girls, Louise Cooper’s Indigo series centers on a very strong female lead.
I see that Glory already recommended The Diary of a Young Girl by Anne Frank. I second that one heartily. It’s not an easy book to read, but I think it should be read by anyone. Girls will find it especially moving, I think, as it’s a very honest book in so many ways.
A series my eldest daughter has recently discovered is The Royal Diaries, wherein each book centers around a different historical princess or queen, from Cleopatra to Eleanor to Jahanara. I like the fact that all the books are about women, and each takes on a different culture and time period. Well worth reading.
For the smaller children:
The Paperbag Princess, byMichael Martchenko and Robert Munsch.
Elizabeth is a princess (about age 12 or so) who lives in a castle and is engaged to prince Ronald. When a Dragon burns down the castle and runs off with Ronald Elizabeth sets off to rescue him wearing only a paper bag (her clothes had been burned up). She confronts the dragon and rescues Ronald whereupon ol’ Ron upbraids Elizabeth for not having nice clothes and smelling of ash and soot (having confronted a dragon). Elizabeth calls him a jerk and the last line is, “And they didn’t get married after all.”
That, my friends, is the best form of strong female character. I once heard a four-year old girl say, “I’m glad they won’t get married. He didn’t treat her right!”
Doesn’t that say it all?
Cynthia Voigt has a number of books with strong female heroines. Homecoming, Dicey’s Song, and Come A Stranger are excellent. The last has an African-American lead character as well.
“Ella Enchanted” is still one of my favorite books, about a little girl who gets the fairy ‘blessing’ of obedience- she has to obey every order given to her- and her journey to try to undo the spell. She’s clever, resourceful, and, when it comes down to it, ethical enough to sacrifice everything she really wants for what she knows is right.
Oh hey, for younger readers, Beverly Cleary has any number of excellent ones (and funny, too)-- the Ramona books are fantastic, as well as others, like Ellen Tebbits and Beezus and Henry.