I wrote myself a documentation when I had to update our instance. Hopefully this helps you out (you’ll have to change the version numbers to what is appropriate for your situation - this is just so you have an idea of the steps you need to do to get to where you want).
Version Naming Convention
(Major).(Minor).(Patch) - ex.: 10.7.3 where 10 is the major version, 7 is the minor version and 3 is the patch version numbers.
Official Documentation
As mentioned in the official GitLab Maintenance Policy, generally you should be able to easily and safely upgrade between patch versions and minor versions within the same major version. HOWEVER, upgrading from a major version to the next major version requires more attention.
The general practice should be to upgrade to the latest available version within your major version. Then proceed to upgrading to the next major version.
For example, when trying to upgrade from 10.7.3 to 11.5.0 - you first need to upgrade to the latest version of 10 which is 10.8.5 and then proceed to upgrading to 11.5.0.
Package Download Locations
All GitLab versions can be located in this package list.
When at 10.7.3
Ubuntu
sudo apt install gitlab-ce=10.8.5-ce.0
RHEL
sudo yum install gitlab-ce-10.8.5-ce.0.el7.x86_64
After the upgrade you might have to run (if GitLab is behaving in a weird way)
sudo gitlab-ctl restart
When at 10.8.5
Ubuntu
sudo apt-install gitlab-ce=11.5.0-ce.0
RHEL
sudo yum install gitlab-ce-11.5.0-ce.0.el7.x86_64
Be a little careful though as people have been reporting issues upgrading to 12.x: GitLab omnibus upgrade from 11.11.2-ce.0 to 12.0.0-ce.0 failed