Start a new game, but don't bother setting up your starting race. This race will end up being deleted anyway.
As you observed, you always start in Sol. From here, turn on Spacemaster mode, open up the System View window, and you should see a button to "Create Habitable". This will create a new solar system with a (more or less) habitable planet. You may not like the system as a whole, so if you want you can generate another system or alter the existing one by adding or subtracting planets, comets, etc or changing their characteristics, all through the same system view window with SM active. From there, select your habitable planet and click "create race". This is when you'll create your actual starting race and their characteristics.
After you've done this, you'll have two player races, the original, throwaway one, and the one you'll actually be playing. With SM mode still active, you can select the Race Window (the one that has the image of a UK flag). At the top left, select your original, throwaway race then click "Delete Race" at the bottom. You can also delete the Sol system itself if you want, in the System View window. From then on you can just play as the race you created in a non-Sol system.
