Update: I recommend to just run the Redmine Docker image instead of doing all this stuff I've written time ago.

$ docker run -d --name some-redmine redmine

Check out this link too: https://hub.docker.com/_/redmine/

The old method

1. Install Apache

$ apt-get install apache2

2. Install rmagick & nokigiri dependencies

$ apt-get install -y build-essential imagemagick libmagickcore-dev libmagickwand-dev libxml2 libxml2-dev libxslt1-dev

3. Install MySQL

$ apt-get update
$ apt-get install -y mysql-client-core-5.5 mysql-server mysql-client libmysqlclient-dev
// enter your top-secret-password

4. Restart MySQL

$ /etc/init.d/mysql stop
$ /etc/init.d/mysql start
$ /usr/bin/mysqld_safe --user=mysql --skip-grant-tables

5. Download Redmine

curl -O http://www.redmine.org/releases/redmine-2.3.4.zip
unzip redmine-2.3.4.zip

6. Change database configuration

$ cp /config/database.yml.example /config/database.yml
$ vim /config/database.yml
// user: redmine
// pass: redminepassword

7. Install RVM + Ruby

$ curl -L https://get.rvm.io | bash -s stable
// logout or source here
$ rvm install 2.1.0 --autolibs=enabled && rvm --fuzzy alias create default 2.1.0

8. Install Pusion Passenger

1. Install Pussion Passenger + Apache mod

   $ apt-get install -y libcurl4-openssl-dev apache2-threaded-dev libapr1-dev libaprutil1-dev
   $ gem install passenger
   $ passenger-install-apache2-module

2. Add the following lines to /etc/apache2/mods-available/passenger.load:

   LoadModule passenger_module /usr/local/rvm/gems/ruby-2.1.0/gems/passenger-4.0.36/buildout/apache2/mod_passenger.so

3. Add the following lines to /etc/apache2/mods-available/passenger.conf:

   <IfModule mod_passenger.c>
     PassengerRoot /usr/local/rvm/gems/ruby-2.1.0/gems/passenger-4.0.36
     PassengerDefaultRuby /usr/local/rvm/gems/ruby-2.1.0/wrappers/ruby
   </IfModule>

4. Enable Passenger as Apache module:

   sudo a2enmod passenger

5. Add a virtual host to your Apache configuration file and set its DocumentRoot to /var/sites/redmine.domain.com/public:

   <VirtualHost *:80>
      ServerName redmine.domain.com
      DocumentRoot /var/sites/redmine.domain.com/public
      <Directory /var/sites/redmine.domain.com/public>
         Require all granted
         Options FollowSymLinks
         AllowOverride none
         Order allow,deny
         Allow from all
         PassengerUser root
      </Directory>
   </VirtualHost>

8. Enable Apache site

$ sudo a2ensite redmine

9. Install dependencies

$ gem install bundler

10. Install Redmine dependencies

$ bundle install

11. Session store secret generation

This step generates a random key used by Rails to encode cookies storing session data thus preventing their tampering.

$ rake generate_secret_token

12. Database schema objects creation

Create the database structure

$ RAILS_ENV=production rake db:migrate

13. File system permissions

The user account running the application must have write permission on the following subdirectories:

1. files (storage of attachments)
2. log (application log file production.log)
3. tmp and tmp/pdf (create these ones if not present, used to generate PDF documents among other things)
4. public/plugin_assets (assets of plugins).

$ mkdir -p tmp tmp/pdf public/plugin_assets
$ sudo chown -R nobody:nogroup files log tmp public/plugin_assets
$ sudo chmod -R 755 files log tmp public/plugin_assets

14. Email configuration (gmail)

$ cp config/configuration.yml.example config/configuration.yml
$ vim config/configuration.yml

Edit configuration.yml

production:
  email_delivery:
    delivery_method: :smtp
    smtp_settings:
      enable_starttls_auto: true
      address: "smtp.gmail.com"
      port: '587'
      domain: "smtp.gmail.com"
      authentication: :plain
      user_name: "redmine.domain@gmail.com"
      password: "secret"

15. Theme (optional)

$ cd public/themes/
$ curl -O http://www.redminecrm.com/license_manager/9722/circle_theme-1_0_1.zip
$ unzip circle_theme-1_0_1.zip

16. Test Redmine

$ ruby script/rails server webrick -e production

Note: Webrick is not suitable for production use, please only use webrick for testing that the installation up to this point is functional.

Fatto!