It is inevitable your Sim family will grow out of their home and need a larger one. The process starts off by evicting your family from their current home. You will have the choice of bulldozing the home. Once your family is evicted, find them in the Create-a-Family screen. After you have chosen to move the family into a lot, you can pick from any of the lots in your price range that don't already have an owner.
When the family moves into their new home, they won't have all their worldly possessions they once had. Instead when you choose to move, the objects on the lot were traded in for cash. The family also gets back the money they paid to buy the lot/house. You will have to start decorating all over again.