Start with the Greeks, specifically Plato and Aristotle
7. Romans: Marcus Aurelius
9. Reverse course a bit: Nietzsche -> Schopenhauer
10. Then read the moderns
11. Then fill in the gaps
I have one comment regarding this, and it is an important one. Personally, I would not sever the direct link between Kant and Schopenhauer, especially pertaining their epistemology. In order to comprehend Schopenahuer fully, one must first have a complete and fresh knowledge of Kant's theories on pure reason. On the other hand, the link between Nietzsche and the greeks is so conspicuous, that one can make the direct jump from Epictetus to Friedrich without difficulty.
But, besides the previous considerations, the advice above given by the redditor cannot be stressed enough...most people involved in philosophy nowadays go for the social and political stuff released on the past 100 years, while completely neglecting Plato, Kant, Schopenhauer and all the others who actually made coherent systems based on imperishable ideas.
More people need to start with a historical guide, and then track ARGUMENTS more than thinkers.
I'd agree with this, except that all people - smart enough to do so, of course - can greatly benefit the development of their own thinking just by perusing the works of the best philosophers, which by their sheer clarity of exposition give a good idea of how they thought what they thought.