In a previous post I suggested reading 'The Last Days of Socrates' by way of preparation for an intellectual holiday to ancient Greece.
To take the actual vacation, why not start off by reading Pierre Hadot's What is Ancient Philosophy?. This will give you a feel for the notion of philosophy as a way of life rather than a mode of theorising about it. The Greeks are particularly interesting to spend some time with because they are both very strange and very familiar - though given the widespread educational neglect of the classics, the 'strangeness' is liable to dominate your first perceptions of them.
You can set your own agenda and timetable. As you dig deeper, by reading works by particular philosophers from time to time (Hadot describes many of the most important ones), you may then sense some affinity. For most of us are emerging from an intellectual tunnel of ideas created by Christianity and only made longer by those great figures, like Marx and Freud, who appeared to be set on dismantling it.
In the writings of the Greeks, we can see a form of thinking about the fundamental problems of life that has not been bent out of shape by the forces within this tunnel. For that reason alone, such thinking can be useful to us - even though we cannot simply transplant it into our own historical situation.
What to pack for the mind-beach.
For a lighthearted general introduction: It's All Greek To Me, Charlotte Higgens
For something more serious that is designed to cater for political correctness and other aspects of modern theorising about identity: The Greeks: A Portrait of Self and Others, Paul Cartledge
As a small child, I found The Greeks by H.D.Kitto endlessly fascinating - though I presumptuously thought I could do better, and wrote to Penguin to tell them so. They replied with a charming hand-written letter saying that although they were sure my book would be very good, they were unable, for legal reasons to offer a contact to someone of such a tender age. I came across The Greeks in a second-hand book shop recently - it has not aged well, but precisely because it did not have to navigate through the mine fields of modern political preconceptions, it throws out some colourful ideas.
Key Works by the two most famous Greek Philosophers
Republic and/or Symposium and The Death of Socrates, Plato
Ethics, Aristotle (I prefer Roger Crisp's translation (Cambridge University Press))
Other books I have found use in this connection
Care of the Self, Michel Foucault
Shame and Necessity, Bernard Williams
The Art of Living, Alexander Nehamas