Both are awesome sports! Generally kids have an easier time starting with nordic then moving to alpine. Nordic skiing has two techniques–skate or classic. Skate skiing is technically more difficult and involves more coordination than classic skiing. Most nordic clubs start out the kids on classic skis. Nordic skiing is also cheap and very accessible to the whole family. It is easier to incorporate it into a daily exercise routine since there are so many more place to nordic ski than alpine. With toddlers and preschoolers, you can even have them practice skiing in your back yard.
Telemark skiing and snowboarding are best learned by grade school kids.
Whichever technique you choose to start your kids out with, remember to keep it fun! Skiing is something your family can enjoy together for a lifetime.