It is recommended to wait until 16.04.1 before updating from 14.04. However I recently updated by servers to 16.04 and didn’t have any issues.

Make sure to update your kernel before updating to 16.04, you can do that by running the commands below.

sudo apt-get update
sudo apt-get upgrade
sudo apt-get dist-upgrade

Once you have done that, run the the command sudo do-release-upgrade. But as always, take a snapshot of the server before the upgrade so you have a method of recovery if it doesn’t work.

I did all of that but sudo do-release-upgrade just gives

Checking for a new Ubuntu release
No new release found

@kurtharnish
do-release-upgrade checks this page: http://changelogs.ubuntu.com/meta-release-lts

That page gets updated 1 week after the point release. Meaning sometime yesterday. You should be able to upgrade today.

BUT:
I just upgraded my 14.04.4 LTS and it went horribly wrong. Many errors on boot and a read only filesystem. Before the upgrade I did a full update/upgrade as @MDS suggests. I had to revert to a snapshot and currently have no idea what went wrong. There where no errors during upgrade, but everything failed after the reboot.

Maybe someone has some suggestions on what I might have done wrong?

In my opinion, you should do a clean install. In the whole history of ubuntu (since 05.10), there are only a very few working upgrades. Especially the upgrade path 12.04 -> 14.04 -> 16.04 doesn’t work (for me). I think, only a fresh installed, otherwise unused system is needed for a working upgrade process. I tried to upgrade several systems (about 10 computers) and every system ended up in a fresh installation. Half of them crashed during do_release-upgrade or upgrade by gui. :-(