Save, or Save As... I use the surname so I can see where I have got up to in my rotation-style play. For Legacy or single-family play, I use a numbering system, eg Gardiner Part 4. It seems it is preferable not to save over old saves too many times as this can lead to corruption. So a date-numbering system can also work.
From playing a family that you want to continue, select Edit Town, then click button that says Change Household.
This lets you select another household and continue playing WITHIN the same Storyline or "Game" as your first family. So their interactions and relationships with the second family will still be effective.
Select the Household you want to play next and click Select.
This is made much easier with later EPs\patches. You can switch households with a new icon on the Family menu.