I'm a big supporter of the Seattle International Film Festival (SIIF). It's a great 3 weeks of films you may never see anywhere else and there is truly something for everyone. Here is a link to their website.
They do have a Films4Families program with some great movies that kids and adults can enjoy. There are some live action as well as animated films this year including what looks like an "awww" film about a white lion cub called White Lion. (The info for these films is on page 20 of the schedule.)
Additionally of interest, there is a Grease Sing-along which could be big fun for those of you with musically inclined children. Another musical offering is the 1916 classic of 20,000 Leagues Under the Sea with ensemble music including the mighty Paramount Theater organ AND author Daniel Handler (aka Lemony Snicket) on the accordion. The other Paramount Theater film is the 1925 silent film Riders of the Purple Sage with music by The Maldives (NW country rock band). The film includes quick-draw cowboys, villains and outlaws, a cattle stampede and an avalanche. (I've attended some silent films with music at the Paramount with my kids when they were younger and it was a fabulous experience.)
They have a FutureWave shorts which are short films by filmmakers 18 and under on Saturday, June 5th. There is also a FutureWave Features as well.
Also, to keep on your radar for next year, are the Youth Juries which are increasingly popular. For the FutureWave films and Films4Families films there are youth juries comprised of 5 students from across Seattle who view these films and then award a Youth Jury Award to their favorite.
And, of course, for the adults, there is Waiting For Superman, about education reform in the U.S. (there are two showings of this; Friday, June 4th and Saturday June 5). I would expect this would get wide release in the U.S. so you will likely get a chance to see it outside the film festival.
SIIF policy for their other films is no babes in arms and the majority of the regular film schedule is not for children (not rated, foreign films, adult material, etc.). Schedules are available at most libraries and many coffee joints and supermarkets. You can buy tickets online, by phone or in person at their box office on the second floor at Pacific Place.