After I copy a "lump" of files to an external HDD or whatev, I just make sure the the number of "files", "folders", and the total "size" are identical ("size on disk" is always different fsr). To make it easy to verify, i just pull up properties of the file copied and the original, and compare them on screen side-by-side.
TeraCopy sounds pretty cool tho, i will have to check it out.
In any case, be sure you verify the new backup of your files before deleting the originals! (this is common sense, but you'd be surprised how being careless for a moment can cause you to lose your files!)