If you want to drag your family to the beach, but want something extra with your trip or a longer stay, head out to Gangwon Province. There are a bunch of things to do in Chuncheon, and there are several areas with pensions near the city, particularly to the southwest along Cheongpyeong Lake. The usual conveniences of a small city - supermarkets, bakeries and pharmacies - are here, and there are options for cycling and other leisure activities by the river.
Day 1: Travel to Chuncheon, stopping off on the way at Petite France - a cluster of French style buildings themed on Antoine de Saint-Exupery's children's book "The Little Prince." Tucked appropriately in the folds of the hills surrounding Cheongpyeong Lake, it's a pretty impressive stab at creating a piece of Europe in Korea.
As an old(er) guy, I like the emphasis on children's activities in the article. There is a lot here that I, a Gangwon resident, would like to do.
If it looks like something you'd want to do later, save it. The Herald has (or maybe had) a policy of archiving material for only a week.