To answer your second question first,
The Virtue of Selfishness by Ayn Rand started me down this path. The sheer brazenness of the title made me pick it up: what I read over the next few days broke me. Or the old me.
I've long left Rand behind, but she's a good start for a bare-knuckle punch to the teeth regarding altruism, egalitarianism and the beliefs of the political Left.
Later I picked up the usual suspects: the Greeks, the Stoics, the modern philosophers. Read the classics of the East and of India, read more modern mystics, read the pragmatists, the existentialists and the Communists. Read the economists (Adam Smith et al.) Read Popper, Wittgenstein, Sartre, Heidegger, Rawls, Kripke, Kuhn, Habermas, etc.
I guess if I were to recommend a starting point, I'd start with Book One of David Hume's A Treatise of Human Nature. Be sure to heed the title of the book--it's a discussion of human nature, not of the world as-is, as-was or as-will-be. And the first book explores how we understand the world (i.e. ideas) and the limits of our understanding. Note it's a little daunting due to the older language (e.g. connexion vs connection, lots of 'tis, 'tho', 'twill, lots of apostrophe-d) and some of his arguments are initially confusing (skip and come back later,) but it will give you a firm grounding on which to consider later philosophy (and much else besides.)
After that, it won't matter too much which books you start with. If you're young, I'd recommend sticking to fiction for a while, pick up some of the old stuff (Homer, Virgil, Ovid, Dante (Inferno,) Rabelais, Cervantes, etc.) Add some new stuff, too (Camus, Tagore, Yoshikawa, Flannery O'Connor, Colson Whitehead, Colum McCann, too many to name.) Once you start hitting thirty or so is when you might pick up some philosophy. The benefit of reading the old stuff first (e.g. Aristotle's Ethics, Plato's Symposium) is that later writers reference them. Even today, were you to read about, for example, the writing craft, teachers still quote Aristotle's Poetics (e.g. all stories have a beginning, middle and end.)
As for personal philosophy, experience will shape you more, especially if/when you have kids, pay higher taxes, buy a house, suffer a sudden medical condition. But understanding the why of it--that's where the philosophers come in handy. Putting your human self--the same human self that is shared by humans from 5,000 years ago--in the context of humanity, of the human community.
(P.S. As a bonus, if you're really itching for philosophy, try out Machiavelli's The Prince as translated by Tim Parks, or for just a little philosophy to start, Rashomon and 17 Other Stories by Akutagawa, esp. "Rashomon" and "In a Bamboo Grove". Another little bit of philosophy can be found in The Master and Margarita by Bulgakov.)
* edit--typos.