Ecuador is not a particularly kid-friendly destination. There are few attractions and activities that appeal to youngsters, and very few hotels here have well-developed kids' programs. But youngsters and teens, especially the adventurous and inquisitive, will do great in Ecuador. The biggest challenge to families traveling with children is travel distances, as well as the logistical difficulties of moving around within the country, which is why I recommend going with two organized-tour options, combined with a stay at a hacienda near Quito.
Day 1: Arrive & Head for Happiness
Alegría means "happiness" in Spanish, and I'd make my family's first stop Hacienda La Alegría. This colonial-era hacienda, still a working farm, specializes in horseback-riding tours, including a comprehensive and conscientious program for teaching beginners and young riders. The hacienda is only an hour or so from Quito's airport, making it possible to come here from all but the latest-arriving flights.
Days 2 & 3: Saddle Up, Partners
Depending on the skills, experience, and abilities of your family, a wide range of horseback rides can be arranged. Beginners will probably stay pretty close to the hacienda, at least on the first day or so. More experienced riders can venture farther afield, through neighboring towns, villages, and countryside, and even up into the high Andean paramo. In addition to riding, kids can take part in various farm chores, including milking cows.
You'll probably want to spend your third night in Quito, because early the following morning you'll be catching a flight to the Galápagos.
Days 4-11: Cruise the Galápagos Islands
The Galápagos Islands are an excellent destination for families. Kids of all ages -- as well as adults -- can't help being awed by the close and constant contact with wildlife. Several cruise operators specialize in family packages to the Galápagos; this is a good way to give your kids some extracurricular class time in the natural sciences. They'll have so much fun, they won't realize how much they're learning. Tauck is a first-rate tour company with several family-oriented itineraries to the Galápagos.
Days 12-14: The Amazon Basin
You can continue your clandestine classes in the natural sciences by following the Galápagos with a 3-day tour to one of the isolated nature lodges in Ecuador's Amazon basin. (To get here, you'll have to fly back to Quito first.) I recommend Kapawi Ecolodge & Reserve, which has excellent guides, comfortable accommodations, and a wide range of tour and activity options. In addition to the chance to see wild caiman, anacondas, bats, and an amazing abundance of bird and insect life, you can take night hikes, and try your hand at fishing for piranhas.
Day 15: Quito
Your return flight to Quito from the Amazon won't get you in until around midday, which is probably too late for your connecting flight back home. If you have time, take everyone to El Telefériqo, a high-speed cable car that will whisk you to the top of Volcán Pichincha. At the base of the cable car is Vulqano Park, where your family can enjoy amusement-park rides and mingle with Quiteño families.