Our family is not really into crowds, so we tend to stay away from the more "popular" spots for holiday times. It doesn't mean I don't know about them, though. If you are in Taiwan over the Chinese New Year and want to show your children some of the culture, then here are some options. I also wrote about a few last year, but I will try to give you some updated links.
Pingxi Lantern Festival
Pingxi is perhaps the best-known traditional spot to enjoy the Lantern Festival and festivities. It can be a long day when factoring in the crowds and transportation, but if you are visiting Taiwan, it may well be worth the energy-sap. There is a comprehensive English-language website already in place, so I won't waste words.
Miaoli Lantern Festival
This year, the main city for Lantern Festival celebrations is Miaoli. It runs from Feb 7 until Feb 28. The official English-language site is pretty comprehensive. This seems to be the common thread of this post! I am pointing to the "well-planned transportation" page only because I like the name so much.
CKS Memorial Hall Lanterns
If you want to take your children to see lanterns, you can always head to CKS Memorial Hall. Here, you will see floats of all descriptions, mostly portraying "The Year of the Rabbit" (this year). It can get pretty crowded, especially at night, but it is very pretty.
I just heard about a special exhibit within the area, called "2011 Lumunarie Festival". (Be ready for the music on that site.) The link is all in Chinese, and I will not speculate about the attraction the event will have for children. There seem to be performances each night after 7pm. It runs from January 27 to February 28 (4pm~10pm). It is closed on Wednesday, Feb. 2, CNY Eve. ($250 for adult, $200 for kids 115~140cm, free for kids under 115cm)
I will add to this as I find out more information, so don't forget to come back!