A sim can move through a variety of ways. Some, however, are unlocked only after progression through the quests.
The three known ways to move from ones home to another in the Sims FreePlay are marriage to another sim, becoming best friends with another sim, or relocating to an empty home for a price. Through the "Love Is In The Air" quest you unlock the option to marry one of your fellow sims. During the objective to move your engaged sims in with each other you also unlock the ability to move your sims who are best friends in with each other.
Relocation is open as soon as you can buy another home.