What is the proper way to style pages in a new installation of a drupal theme?
Both or either are just fine, it really depends on what you want to achieve.

Panels will give you a UI in which you can move sections of the page around, but you will still need template files for the layouts if you don't want to use the standard ones.

Rearranging a template file is fine, but much less extendable via the UI.

In short, it's a matter of choice.  I recommend trying both and seeing which fits your needs best.

Also, check out Omega 4.x, it's a beast of a theme with some extra layouts built in and a pre-built gemfile so that you can use SASS if you want.