Open the service and choose Upgrade/Downgrade. The page shows the plan you are currently on and the plans you may move to; pick one and press Request change.
What happens next depends on the direction. An upgrade is priced for the remainder of the current period - you pay the difference, not a full new term. The service's resources change once the request is processed and any difference is settled.
Your files, databases, email and apps ride along untouched: a plan change resizes the account, it does not rebuild it. Downgrading to a plan smaller than what you currently use - more disk in use than the new plan allows, more apps than it permits - is the one case to check before requesting.
If the page says no changes are available, the operator has not defined an upgrade path for your product; a ticket is the way to ask.