I understand that hybridisation occurs due to the atom wanting to exist in the most stable state possible, and the most stable state has the lowest energy. What I don't understand, however, is say, in CH4, the bond angle is 109°18', so how does this particular value mean that the atoms and bonds in methane are farthest from each other? And if they are indeed farthest from each other, how does that mean the CH4 molecule's in the lowest energy state?
VSEPR stands for Valence Shell Electron Pair Repulsion. It's a complicated acronym, but it means something that's not difficult to understand. Basically, the idea is that covalent bonds and lone pair electrons like to stay as far apart from each other as possible under all conditions (to minimize electrostatic repulsions of different pairs of electrons) . This is because covalent bonds consist of electrons, and electrons don't like to hang around next to each other much because they have the same charge. This, of course, allows the molecule to be in its lowest energy state.
This VSEPR thing explains why molecules have their shapes. If carbon has four atoms stuck to it (as in methane), these four atoms want to get as far away from each other as they can. This isn't because the atoms necessarily hate each other, it's because the electrons in the bonds hate each other. That's the idea behind VSEPR.
Now, one problem with the whole VSEPR thing is that if you have four things stuck to carbon, for example, there are no orbitals that want to get 109.5 degrees apart from each other (109.5 degrees corresponds to the geometric maximum distance the atoms can get apart). After all, s-orbitals go in a complete sphere (360 degrees) and p-orbitals are 90 degrees apart.
What happens instead of using s- or p- orbitals is that when covalent bonds are formed, the s- and p- orbitals mix to form something called hybrid orbitals. "Hybrid" just means "mixture of two different things", and that's exactly what a hybrid orbital is. When three p-orbitals with 90 degree angles combine with one s-orbital with 360 degrees, they average to form four sp3 orbitals with 109.5 degree bond angles. Depending on the numbers of s- and p-orbitals that mix, you can get a bunch of different bond angles.