Mid 2010 Mac Pro 5.1 Drive Is Failing It still works but am afraid it will unexpectedly die. I bought a new drive that's compatible but don't know how to transfer data from old to new after I install the new drive. 

Can I use Time Machine, Disk Utilities or what?

The old drive is Sierra 10.13.4 and the new drive is formatted with Sierra 10.12.
First (after installing the new HD) you'll need to install macOS on the new drive. You can use internet recovery (boot with CMD+OPT+R) or make an install USB.

If you have an external hard drive available you can use Time Machine to create a back up and then restore that backup to your new machine.

If you don't have an external hard drive you could upload your files to iCloud or another online service and restore them later.

Another option is to purchase a SATA to USB adapter (such as this) and either use Migration Assistant or simply copy the files you'd like manually.