I am planning a family (2 teenagers) trip 3 days in mid July. We hope to spend most of our time exploring the mountains on day hikes. What area would provide a better home base Canazei or Cortina d'Ampezzo
My choice would be Canazei. Will you have a car? Take the Sass Pordoi cable car and go hiking up there. Pretty amazing. Nothing wrong with Cortina. It's a little more upscale and has a lot of hotels and perhaps things to do, but the Sella Massif is more to my liking if outdoor activities are your aim. Hotel Astoria in Canazei is top end. Very good food if you take the half board option.
You might want to consider one of the smaller villages such as Kastelruth or St. Ulrich...you can take cable cars from both of them and be in the mountains and up on the beautiful alps in a matter of minutes. Cortina is OK but it's a bit of a scene...lots of Rome license plates and people parading their $$$...reminded me a little of Aspen...not my favorite place but beautiful hiking directly out of town. Bruneck is also nice as well as Toblach...smaller, very authentic places that we love with hiking all around. You really can't go wrong anywhere in the Dolomites especially if you'll have access to a car.