If you want to use HTTPS you have to setup your Apache to do so. There are many articles describing how can you do it on Ubuntu. Try searching "ubuntu setup ssl apache2" or so.
Example: http://www.linode.com/wiki/index.php/Apache2_SSL_in_Ubuntu
In few words: you have to create a new NameVirtualHost entry in your Apache configuration, and make it listen to port 443 (default https port) and setup a certification. If your website is a public website then I suppose you don't want a self signed certification (the browser will tell the user that the certification is signed by the server itself, therefore it cannot be trusted). Look for an SSL certification on the net, I have used Verisign for one of my projects. There shouldn't be any big differences between certifications, users normally not aware of what is happening under the hood, although I would suggest to buy it from a more well known name (users might like it better).
Symfony is not responsible handling your requests on HTTPS level, if your webserver is configured properly, the request comes through HTTPS and it will load PHP and Symfony to generate the response and send it back through HTTPS. What Symfony can do for you is to redirect pages between the http to the https site, if you wanted to (or wanted to limit some pages, like everything is unsecure except the login, register and change password pages). Unfortunately I don't know any good plugin to solve this, but basically you have to write a filter that triggers on any request, just before the request handling module/action is being processed.
This source looks to be good: http://blog.tigregroup.com/programming/creating-an-ssl-redirect-filter-in-symfony/
hth