You cannot transfer primary calendars to another user, so open your events and copy them to a new (or existing) secondary calendar. This will preserve invited guests. If you don't have guests on your events, or it's just historical events you could simply Export and re-import all of your calendars via .ical (export won't save the "guest" statuses like transferring will).
If it is a secondary calendar (or if you just moved your upcoming events there), invite your other account to share the calendar, then unsubscribe the original owner from the calendar.
Contacts makes it very easy to export data. Visit the main contacts
page directly, or within Gmail. At the right near the top are links
called Import, Export, and Print. Click the Import link to initiate
the import process.
Update: Aug'13: It is now possible to transfer Ownership of a Document in Google Drive.
The only way is to either upload the files again to the Google Apps account or share the files with the Google Apps email address of yours. You cannot transfer the Ownership of the files yet (if you have upgraded Google Docs to Google Drive)- this feature will be coming shortly. If you are still using Google Docs, the owner can be changed to the Google Apps Account.
I had a lot of Google documents to migrate from a free account to a Google Apps academic account. The best way I found was using the Google Drive Desktop (PC or Mac) to make the copies. Here are the steps:
Google Drive Web:
In the free account, share the documents with the Google Apps academic user
In the academic user account, move the "Shared with me" files/collections into your main list (or use the Add to My Drive button). More info To make it easy, you can create a collection called "Files to migrate" and organize the things in there. This is so the shared stuff from the free account will show up on your PC/Mac.