What's the best way to do it? What module(s) should I use to export the data from the Drupal 6 site and import the data into Drupal 7 and what do I need to think about regarding the different branches? Please also describe the process if you have experience with it.
First, upgrade the site to Advanced Forum 6.2.x, then, after everything is correct, migrate to D7.
The upgrade to 6.2.x should be relatively smooth (there were no problems in my experience).
You might have to have several tries for the D6->D7 migation, depending on your site configuration. Always do it in a separate installation, and back up everything, step by step. If something does not go right, this will allow you to reset the site to a point where you can restart a step, rather than going back to a fresh install and starting from scratch.
Follow the Upgrading from previous versions on D.O. to the letter.