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I'm new to differential geometry, and I haven't seen any concrete examples in it yet, so I do not yet know how to show things there. I've read that the sphere $S^2$, for example, is mapped to $\mathbb{R}^2$ via stereographic projection by removing one point (say, the north pole). Does this mean that one of the charts in the atlas of $S^2$ is the stereographic projection $\phi: S^2\to \mathbb{R}^2$, and the other one is $\psi: (0,0,1)\to \infty $?

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Not quite, remember that charts are homoemorphisms (or in your case diffeomorphisms) with $\Bbb R^n$ and more importantly: they can overlap. So one example--in this case--for the second chart is stereographic projection off of the south pole. You can do the same thing for general spheres as stereographic projection is a thing for all of them.

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    Do these two stereographic projections kind of map to $\mathbb{R}^2$ to both sides of the "sheet"? – 2017-02-06
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    @sequence don't think of them as a "sheet" or even as the same thing. The idea behind charts is that they are abstract isomorphisms (again in your case this means diffeomorphisms) with $\Bbb R^2$. Stereographic projection as an "action" is just a way to visualize how the map is constructed. You make them **separately** then put them on $S^2$ to cover it up. – 2017-02-06
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    But if we have $S^2$ with $(0,0,1)$ pointing to infinity, then if we create a stereographic projection from $S^2$ to $\mathbb{R}^2$ with $(0,0,-1)$ removed, then $(0,0,1)$ will still point to infinity. Can you please clarify? @AdamHughes – 2017-02-06
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    @sequence no, you project from the point you remove, so $(0,0,1)$ goes to $0$, not $\infty$. – 2017-02-06