Plug in the new one, then copy them over in the data management screen in the settings menu. Do NOT use the move function, because if a move fails halfway through, it will delete the source files as well, and you'll lose your files on the old drive.
Once they are copied over to the new drive, you can unplug the old drive while in the data management screen, as long as you aren't on the screen showing the old drive. Do not pull out the old drive while on the System Menu or the Wii U will crash.