How do i sign into a different iCloud account on my iPhone?

If you just want to use a different iCloud account for Mail, Calendars, Reminders Contacts (without App Data & Documents), then just set it up in "Settings > Mail, Contacts, Calendars > Add Account... > iCloud" otherwise (if you want to use all iCloud features with your other account) then you will have to delete your former account and add your new account in "Settings > iCloud".

Settings>icloud, scroll to bottom of screen and tap Delete Account. Then enter the account that you want to change to. This does not delete any data on icloud.

what if you need the data already on the iphone to stay in the cloud, just deleted from the iphone?