Energy of electrons in the same subshell
It's important to remember (or to learn, if it hasn't been explained before) that the only atomic systems for which we can resolve the Schrödiger equation exactly are what we call hydrogenoid atoms - atoms with only one electron, such as $\ce{H}$, $\ce{He+}$, $\ce{Li^2+}$...

As soon as you add a second electron, atomic orbitals change and they aren't exactly the same as for the hydrogen atom (this is a form of the three-body problem that also comes up in other fields of physics; if you remember your Newtonian mechanics, that's why you cannot solve analytically a three-body gravitational problem). The source of this change is electron-electron interaction.

However, electron-electron interaction, in general, introduces only minor distortions in atomic orbitals - for the most part, the orbitals of poly-electronic atoms are very very similar to hydrogenoid atoms in terms of shape, size, etc, which is why we use the same labels for them as for hydrogenoid atoms ($\mathrm{1s}$, $\mathrm{2s}$, $\mathrm{2p}$, etc).

There is one property of hydrogenoid atoms that is very importantly changed by electron-electron interactions, though: in hydrogenoid atoms, the energy of an orbital is only a function of its main quantum number, $\mathrm{n}$, which means that all the orbitals of the same shell will have the same energy. In a single-electron atom, an electron in a $\mathrm{2s}$ orbital will have exactly the same energy as an electron in a $\mathrm{2p}$ orbital, because they only interact with the nucleus through a Coulombic interaction - which has spherical symmetry (it's the same in all the directions of space; it only changes with distance to the nucleus).

If there are other electrons in the same atom, occupying different orbitals, this breaks down the energy equivalence of same-shell orbitals - because, for instance, an electron in a $\mathrm{1s}$ orbital interacts differently with an electron in a $\mathrm{2s}$ orbital or in a $\mathrm{2p}$ orbital - due to symmetry, overlap, and a number of factors (these interactions are very complex, can't be solved analytically, and to some extent aren't even completely understood). The energy corrections are generally small, but the orbitals belonging to the same shell are no longer equal in energy - that's why we can order subshells, and we know that $\mathrm{2s}$ orbitals fill up before $\mathrm{2p}$ orbitals.