I am trying to upgrade my Ubuntu 14.04 to Ubuntu 16.04. My update manager notifies me that a newer version is available. It also gives me the upgrade button.
When I click 'upgrade', it asks for password, but after that, literally nothing happens.
I tried this several times with no luck, and there doesn't seem to be a similar issue reported anywhere.
Try running
sudo do-release-upgrade
If it fails, update your question to include the output.
Online upgrades can be done and I've done quite a few successfully over the years but in many instances they fail*. Also note that even when successful, the whole process is tedious (it can take hours), and prone to errors because it may require user intervention more than once to answer a few questions that more often than not are quite daunting for new users.
Assuming your have done your backups, I strongly advice a clean install instead.