If there is not already any data on the SSD at all, then it is pretty easy: boot the existing install in rescue mode ( choose from the grub boot menu ), and then at the rescue menu, choose the root shell option, then run
lsblk to see what disks are what. Identify which one is the existing USB, and which one is your new SSD, and assuming the USB is /dev/sda, and the SSD is /dev/sdb ( adjust the parameters otherwise ), run
dd if=/dev/sda of=/dev/sdb to copy everything on the USB to the SSD. You should then be able to shutdown, remove the usb, and boot the SSD and fire up gparted to move/resize your partitions to take advantage of the additional space. Don't try to boot with both disks connected at the same time, as they will have the same UUIDs and this will confuse the system.
If you have data on the SSD already, then it is slightly more complicated. First, create a partition on the SSD to hold your root fs. You don't need to format it. Then boot into rescue mode as above, only instead of copying the entire disk, you need to copy only your root partition. Assuming /dev/sda1 is the existing root, and /dev/sdb3 is the partition on the SSD you want to copy to, run
e2image -arp /dev/sda1 /dev/sdb3 to copy the filesystem ( unlike
e2image is smart enough to copy only the used parts of the filesystem, and skip the free space, speeding up the process ). After that, run
resize2fs /dev/sdb3 to expand the filesystem to use the additional space in the new partition. If /dev/sdb4 is your new swap partition, format it with
mkswap -U uuidgoeshere /dev/sdb4. Use the UUID of the old swap partition. Finally, install grub on the SSD so it can boot:
grub-install /dev/sdb. After that, shut down, remove the usb stick, and boot from the ssd.