It's possible to back up your contacts to your SIM card before transferring it, but this isn't always the most convenient option. It's useful if you're moving to a dumb phone, but if you're using smartphones your contacts are typically synced with your Google account or Apple ID.

How would you rephrase that briefly using Chinese?
打开旧手机的联系人列表。 打开菜单并选择“导出”或类似选项。 选择把联系人导出到SIM卡上。