Urgent for work, it was my old work computer and need all the files on my old hard drive to put into an external storage device and then use that to put the files into a new computer or is there another easier way to get the files off my old hard drive.
That's probably the easiest way. Get a usb external drive, copy from your old computer. Plug the external drive into your new computer and copy files to the new computer.
The other way is to put your files on a network share that hopefully your IT dept. provides that will be backed up so you won't lose your files and they will be available from any machine on the network.