Moving boot drive to larger hard drive

Is it possible to take my Mac boot drive that's installed on a 750GB hard drive and just copy it over to a 2TB drive so that I can have some extra space?

I tried copying the image but it ended up formatting the 2TB drive down to 750GB.

As you discovered copying the image doesn't quite produce the result you want. There are several programs available that can "clone" your original drive to the new 2 TB. You can do this using programs such as Super Duper or Carbon Copy Cloner. Here's how to do it using Disk Utility (assuming you have a recent version).

1. Remove any data that you want to save from the 2 TB drive.
2. Launch Disk Utility and select the 2 TB drive on the left. Click the Erase tab and format the drive as Mac Extended Journaled.
3. If you are going to partition the drive do that now. Remember that Intel Macs need to boot from a GUID partition.
4. Click the Restore Tab. Drag your current boot drive from the left to the "Source" blank. Drag the 2 TB drive to the "Destination" field. Then clic Restore.