As a general rule, newer minor releases (e.g. v14.1 etc) are of most value to new users. To upgrade most of the operating system, you can just do an "apt-get update && apt-get upgrade" and that will bring all the packages up to date (actually, many of them will be newer than v14.1).

Alternatively you could use TKLBAM to migrate your existing install

But that won't update the included Redmine version itself. You'll still need to do that manually. Although, from what I've read, it should be fairly straight forward. The Redmine upgrade instructions are here.

The only thing worth double checking is that you have a sufficient Rails version. Ruby itself should be fine as Debian Jessie has Ruby 2.1.5 and according to the requirements it supports 1.9.33, 2.0.02, 2.1, 2.21 & 2.3. AFAIK, our appliance has Rails v4.2 installed via gem (Ruby package manager) so you should be good there too (Redmine requires Rails v4.2). So it should (at least in theory) be pretty straight forward.

When you get to Step 3, follow Option 1 (download). Otherwise, it should be just a case of following the instructions...

I'm not a big fan of doing this sort of thing on a production server (just in case something goes wrong), so personally I'd use TKLBAM to migrate your existing install to a VM. Then upgrade Redmine. Please note, that even if you do migrate your data to the newer version, you'll still need to upgrade Redmine. This is for 2 reasons.

First is that Redmine will still be out of date in the latest TurnKey version (it was released quite a while ago, we're not too far away from the v14.2 release). Second, is that your backup actually includes all the Redmin software as well. So even if you were restoring to the very latest version (with the latest version of Redmine pre-installed) it would overwrite that with your data (including the Redmine software).

So personally how I'd do this as as follows:

if you don't have a recent tklbam backup, create one

launch the v14.1 Redmine appliance as a VM (e.g. using VirtualBox or whatever you have handy)

Restore your backup to the VM

Go through the Redmine upgrade steps (no backup required, because you're already working with a backup, not your production server)

document as you go, so worst case, something goes wrong, you can rinse and repeat without needing to remember what you did

once everything is working as it should, you have 2 options:

1. repeat the steps you documented on your production server; or
2. simply create a backup of your new server and restore it back to your production server

Please be aware, that if you choose option 2; any changes to the production server between your most recent backup (that you transferred to the VM server) and when you restore, will be lost!

Also IMO this demonstrates, the value of using VMs in production. If you were using some sort of hypervisor (even if it only has one VM running on it) then you could simply decommission your production server and keep the new VM (rather than having to go back and repeat yourself).

I hope that helps.