Apart from Will Durant's profundity as a historian I think most of his devotees would agree that he was an artist with the pen. Durant's works are very readable and suggest that throughout his writing career both clarity and consideration for the reader's efforts were concomitant in his mind with the task of historical and/or philosophical instruction. Will wrote for 'the rest of us' and made the dusty and often obfuscated paths to knowledge painlessly accessible to the general public.
The only books that could be called "companion books" (albeit loosely) that I can think of are: 'Heros Of History' and 'The Lessons Of History', but really ... any volume of 'The Story Of Civilization' can be opened at any page and read on its own merits - without any accompanying text.
I would suggest simply starting at the beginning with volume 1 (Our Oriental Heritage) and continuing; however, perhaps one could choose a period of history of which one has greater interest as a literary appetizer if-you-will. In either event you are about to embark on a magical journey of discovery piloted by a master writer and storyteller. Bon voyage, and send us a postcard ! ;-)