The main instrument of the spread of Greek culture was in establishing colonies abroad. The main sites of these were in Sicily and southern Italy (eg., Naples, Taranto, Syracuse), but throughout the Mediterranean, including Libya, France (Marseilles), Asia Minor (eg., Smyrna, Chalcedon, Byzantium) and Spain. If you look up 'Colonies in antiquity', it will be easy to find a map for this.
Secondary to the above ways to spread in Greek culture was trade, and the imperialism of Alexander the Great.