Will syncing to a new computer erase everything on the iPhone?

Not everything - just as provided by the warning message. All iTunes content will be removed from your iPhone with the first sync for iTunes content with iTunes on a new computer - unless you transfer your iTunes library from your old or existing computer to your new computer per the instructions included with this link first. All apps on your iPhone should be in your iTunes library on your computer.

http://support.apple.com/kb/HT1751

The Camera Roll is never touched during the iTunes sync process and the same for contacts under you select to sync contacts with a supported address book application on the computer, which must be manually selected under the Info tab for your iPhone sync preferences with Tunes.

Photos/videos in the Camera Roll can and should be imported by your computer as with any other digital camera. With a Mac, you can use iPhoto for this, or the Image Capture application.

Before syncing contacts with an empty Address Book and calendar events with an empty iCal on the Mac, enter one contact in the Address Book and one calendar event in iCal. This will provide a merge prompt with the first sync for this data, which you want to select.