Okay...first a caveat. I'm 39 as of today and my partner (36) and I do not have children.
That said: children should certainly enjoy -
Future World -
1. the Living Seas with Nemo (and Crush's Turtle Talk)
2. Test Track (assuming they are tall enough)
3. the Land - the Circle of Life movie and Soarin' (depending on your children, Listen to the Land may also be interesting at least once).
4. the Figment ride (I'm not sure how they would enjoy the movie...the characters are from a movie from nearly 20 years ago, so they wouldn't be familiar with them AND the effects are intense. If you must do a 3D movie on property, I'd recommend Philharmagic and Muppets FIRST (in that order)).
5. Innoventions has all kinds of hands on exhibits in both buildings
6. the Character Connection features character greetings throughout the day.
World Showcase -
Tour the world in Counter Clockwise fashion so they don't ride El Rio Del Tiempo in Mexico and Maelstrom in Norway FIRST but LAST...that way they won't be conditioned to EXPECT a ride in every country.
1. Before touring World Showcase, buy a pair of passport sets. They include stickers from each country and a space for a cast member in each country to write something to the kids. They may want to accomplish something in each country before filling out their passport (to make it a game). Adults might choose drinking around the world

But, for the kids maybe buying a postcard of each country or buying some candy in each country or simply taking a photograph in each country (maybe finding a hidden Mickey in each country?). There are Kidcot benches at every country where they can design a mask as they progress around the lagoon.
2. in Canada: I really enjoy Off Kilter (the Celtic Rock band). You may also see Koda from Brother Bear here.
3. in UK: The World Showcase Players put on fun (partially improv) sketches with guests in some rolls, there are also concerts in the back of the pavillion by the British Invasion (a Beatles cover band) and characters from both Winnie the Pooh and Mary Poppins make appearances throughout the day (and sometimes Alice in Wonderland as well). The Cadbury chocolates here are amongst my favorites: try a "curly wurly" (sort of like the late lamented Marathon bar), a "double decker" bar or a "picnic". Other friends of mine also enjoy a "flake" bar.
4. in France, you may find the juggling waiter, fresh baked pastries and Belle, Beast and Aurora.
5. In Morrocco, you might find Aladdin, Genie and Jasmine, fun shopping and if you may be a little adventurous about your menu, try eating here for lunch or dinner in the Restaurant Marrakesh...your children will be invited to dance with the belly dancer! Oh! Don't miss the acrobats here or the band Mo'rockin!
6. in Japan, you'll find the single best shopping place (in my opinion), including interesting Japanese snacks (and manga/Pokemon/Yu-Gi-Oh type stuff). I recommend: Pocky, Botan rice candy and Morinaga Hi-Chew candy. You'll also find the museum, which last featured an exhibit on Japanese Tin Toys of the 50's and 60's. Also fascinating in the store is the "buy an oyster/pick a pearl" tank where a jeweler will set your own pearl for you (admittedly not as interesting to the kids, but they may be interested in watching). Miyuki, the candy carver/molder is awesome and makes the most fabulous creations out of taffy. I also enjoy the drummers here!
7.in American Adventure - well, the kids may not find the show interesting until they are a little older, but don't miss the funnel cake!!
8. Italy has some nice candy in the store, but I don't really know about other kid friendly stuff here (no attractions or performers jump into my mind), other than Pinocchio and Geppetto making occasional appearances here.
9. in Germany, be very careful 'cause a great deal of the one shop is crystal and Hummel figurines...maybe bypass Germany too...well...except for the miniature train set which the kids love! Snow White and the Dwarves do make appearances here.
10. at the African Outpost, the storytellers/drummers always draw a crowd of kids. This is where you might also find Timon or Rafiki.
11. in China, the kids will love to watch the acrobats (many of whom are younger kids) and the shopping here is also very fun with lots of little cheaper things that the kids may be able to afford. Mulan and Mushu make occasional appearances here.
12. in Norway, there is a viking ship shaped small playground for the kids to blow off steam, there is Maelstrom (which has always been a little short for me, but the kids may enjoy it), the bakery has great snacks, too!
13. in Mexico, you'll find El Rio Del Tiempo, more fun shopping, and there is a "Children's Please Touch Museum" type of exhibit on a typical Mexican home. Somewhere in there, you can also send home a video of yourself (via e-mail) "cliff diving", etc.
Also, Illuminations is a wonderful show (though I personally rank it second behind Wishes for end of night shows).
On your way out of the park...check out the illuminated fiber-optic lights on the ground near Spaceship Earth!