If you're just getting in to it, the best way to get a sense of Ancient Greece is probably the twin introductory books published by Wiley-Blackwell - Jonathan Hall's A History of the Archaic Greek World (2007) and P.J. Rhodes' A History of the Classical Greek World (2006). These books are designed to cover all the major topics, from a basic timeline of political events to society, political systems, culture, art, warfare, and so on. Another really good introduction, compiled by a group of absolutely world-class scholars, is Robin Osborne's Classical Greece, 500-323 BC (2000). From these works you can get a sense of what topics interest you in particular.

There are a lot of meatier introductory textbooks out there, but if you're coming to the topic with little prior knowledge, it's usually best to build it up slowly. Once you've digested the basics, you can get a more critical and scholarly approach to the period from Robin Osborne's Greece in the Making, 1200-479 BC (1996) and Simon Hornblower's The Greek World 479-323 BC (4th ed. 2011).

I wouldn't recommend starting with the primary sources; it's better to read some broader treatments first, so that when you turn to ancient texts you know roughly where they fit (in terms of their authors, perspectives, genres, and the things they are describing). However, once you have some background, reading the sources is definitely the best way to get a sense of the Greeks themselves. Homer and Herodotos remain fan favourites; Thucydides is perhaps the most highly regarded historical source of all of Antiquity; Xenophon does more to make the Greeks seem human than any number of wordy philosophers.