I agree with Mauro ALLEGRANZA. You can either dive right into the last 100 years or so of the conversation since the participants speak your language and are talking about topics that you care about, or you can take a year or two to dive in and catch up on the conversation's beginning and middle.
I would start with a smattering of books (say, 4 per year) in the history of philosophy, using a solid list like St. John's or Torrey Honors. Pick one or two books from each semester and work your way through.
You'll be surprised at how exactly the terms and ideas align to the present day conversation and inform its presuppositions, goals, and entrenched disagreements.
So for instance, a 3+ year reading list might include:
Unit I Ancient Thought
- Plato, Gorgias
- Aeschylus, Agamemnon
- Aristotle, Nichomachean Ethics
- Sophecles, Oedipus
Unit II Medieval
- Bible, Genesis
- Lucretius, De Rerum Natura
- Aquinas, Summa Theolgiae (the questions listed by St John's)
- Dante, Divine Comedy
Unit III Early Modern
- Machiavelli, the Prince
- Bacon, Nobum Organum [selections]
- Descartes, Meditations
- Pascal, Pensees
- Hobbes, Leviathan
- Hume, Treatise of Human Nature
- Tocqueville, Democracy in America [selections]
Unit V Modern
- Rousseau, Discourse on the Origins of Inequality
- Adam Smith, Wealth of Nations
- Federalist Papers [selections]
- Supreme Court Cases (pick a few that interest you)
Univ IV Contemporary (a VERY wide scattershot of views)
* John Rawls, Justice
* Noam Chomsky, American Power and New Mandarins
* Thomas Nagel, Equality and Partiality
* Nicholas Wolterstorff, Justice: Rights and Wrongs
Depending on your reading pace, you could read 4 every 6 months (and do two units a year, for 3 years) or 4 a year for six years. As a professional, either might be realistic.
As you read, you will almost certainly need someone to dialogue with, in person or online. Also, you will almost certainly come up with a list of other books you'd like to read when you finish (books that are continually referenced, like Plato's Republic). But try to stick to the list and see the whole story/conversation unfold into modern times. You can't read everything at once. Finish, then build a new list.
Hope that helps!