Go to Settings > Reminders, where you can:
Set a default list for new reminders
Sync past reminders
Keep your reminders up to date on other devices. Go to Settings > iCloud and turn on Reminders. To keep up to date with Reminders on OS X, turn on iCloud on your Mac too. Some other types of accounts, such as Exchange, also support Reminders. Go to Settings > Mail, Contacts, Calendars and turn on Reminders for the accounts you want to use.