I have an old laptop with Ubuntu 15.x and bought a new SSD to speedup the system a little bit. To copy the whole system from the HDD to the SSD you need to align the new partition on sector numbers which are dividable by 4096. So I have to create a primary partition starting on sector 4096 of the SSD and a Linux Swap partition. Use gparted and check the sector numbers twice.

To copy the existing data I found an article, which uses rsync:

rsync -rvlpogdstHEAX /hdd/ /ssd

It copies the Grub too, but in my case, the Grub uses UUIDs, which differ between HDD and SSD. So I have to change the grub.cfg manually:

chmod 644 /boot/grub/grub.cfg
vim /boot/grub/grub.cfg

Use now the vim replace function:

:%s/old-uuid/new-uuid/g

It replaces all occurrences of old-uuid with new-uuid. Then you need to open /etc/fstab and replace the UUIDs for the primary partition and the Swap partition too.

Install the SSD on the computer and start it, it boots your old system.