At the bottom of most App Store tabs you will see your Apple ID in a button. Tap that to sign out of your business account, then tap it again and sign into your private account. Once you do that you will be able to visit the purchased tab in the store and download your apps.
You'll have to be careful to remember what apps are on what account, because to update them you will have to keep switching back and forth between the accounts. The iTunes and iBook Store logins are separate, so each store will need to be logged in and out of individually, which in your case is probably more convenient than not; since you'll probably only need to change the App Store more than once.
For calendar information, if you are not already syncing with Google, iCloud, or some other online service, I recommend you do that first. Then you can simply remove the account from the old iPad (or restore the iPad), and add the account the new iPad, which will then sync.
Enjoy your new iPad.