If you love shopping then don't expect too much from your holiday in La Gomera. Perhaps the most fun shopping is to be had in the supermarkets picking out interesting and unusual food that could be gifts for the folks back home.
The island's capital San Sebastian has the best shopping on the island, but it's not extensive. It's a pleasant place to wander though, and on a good day you get great views over to the island of Tenerife. Maybe pick up a bottle of local wine, made from Malavasía grapes.
Mountain bikes are available in most of the bigger towns, and can be a fun way for the family to get around towns, but don't think of going out on the more hazardous mountain roads.
Alternatively take a drive up into the mountains to the highest point on the island: the top of the Garanjay is an easy 20 minute walk from the car park. You are likely to see some huge waterfalls during the trip too. Kids might be impressed by the wildlife such as the little lizards sunning themselves on the rocks.