Is there any specific upgrade path to follow?
The official recommendation is not to skip major versions. That is follow something like 8.x => 9.x => 10.x. Check out https://docs.gitlab.com/omnibus/update/README.html for details.
Are there any caveats in directly upgrading it from 8.x to 10.x?
Yes. There are some deprecated configurations and version changes in that path. Check https://docs.gitlab.com/omnibus/update/README.html#updating-from-gitlab-8-10-and-lower-to-8-11-or-newer
We are using a 2GB/2core DO droplet with ubuntu 14.04, psql=V9.5
The recommended RAM is 4GB (including swap). Please check https://docs.gitlab.com/ce/install/requirements.html for specifics. Also, you will need to upgrade PostgreSQL to version 9.6. Version 9.5 is no longer supported (as a matter of fact, version 10 is also not supported yet. So bump to 9.6.x)
Also how to find if we are using omnibus installer ? (Need to find how previous admin insatlled it)
See if the package
gitlab-ee have been installed. Those are omnibus based packages. (You can also check if there is a command
gitlab-ctl available. Source installations will not have it)