The procedure is just a matter of logging in and changing the plan. So long as you're not in a trial period, you're able to switch.
Dropbox is included in the Pro plan.
You can keep your existing Dropbox subscription, or cancel it. If it's an annual subscription, you'll receive a pro-rated refund on that from Dropbox.
If you keep the existing (personal) Dropbox account, the data stays there. If you cancel it, your data migrates over seamlessly, no re-uploading required.