> After I want to install a new Ubuntu to the notebook, how can I copy...
I think you mean _first_ copy the files to an external drive, _then_ reinstall Ubuntu.
You can't copy files with the GRUB console: the filesystem drivers of GRUB are basically read-only. (You can write into e.g. the `/boot/grub/grubenv` file, but only by overwriting its existing contents - you cannot increase the size of the file.)
With the `initramfs` console, you would first mount the necessary filesystems (= at least the filesystem you want to copy _from_ and the filesystem of the external hard drive you want to copy _to_.), and then use the regular `cp` command to copy the files.
But there is probably a better option: prepare an Ubuntu Live DVD/USB, boot from it, and then you have all the basic filesystem manipulation tools (even GUI!) available to you.