What is the best way to use Panels with the Omega theme?
If you want to use Panels, Omega is probably not the way to go.  The lead Omega developer is a big supporter of the Context module (as noted by Letharion) and you can see an example of the slow uptake of panels support on this thread.

If you're new to Drupal and want to use Panels, I would recommend trying Adaptivetheme, especially the 3.x branch (currently rc1 but due for a stable release soon).  Adaptivetheme has a long history in the Drupal community and has nearly 30,000 sites using it; it also has the best Panels support of all the major theme frameworks.  The 3.x version of Adaptivetheme adds support for responsive design in Panels, easy CSS resets, etc., making it very comparable to the Omega theme but allowing for excellent Panels support.  There are also a variety of subthemes available (links on the module page) that should give you a good idea of what this theme is capable of.