How are forces "mediated"? I hope this is the right word to use.
To me, these forces seem kind of fanciful (except for General Relativity and Gravity, which have a geometric interpretation).
For example, how do two charged particles know that they are to move apart from each other?
Do they communicate with each other somehow through some means?
I've heard some people tell me that they bounce together messenger photons.  So does one electron receive a messenger photon, go, "Oh hey, I should move in the direction opposite of where this came from, due to the data in it", and then move?
Aren't photons also associated with energy, as well?  Does this type of mediation imply that electrons give off energy in order to exert force on other electrons?
Every electron is repelled by every other electron in the universe, right?  How does it know where to send its force mediators?  Does it just know what direction to point it in?  Does it simply send it in all directions in a continuum?  Does that mean it's always giving off photons/energy?
I'm just not sure how to view "how" it is that electrons know they are to move away from each other.
These questions have always bugged me when studying forces.  I'm sure the Standard Model has something to shed some light on it.
 A: The short answer in quantum field theories is "by the exchange of virtual particles".  Look up "virtual particles" on Wikipedia to get a sense of it.  The ancillary questions you asked about the energy budget, and the derivation of Coulomb's law are very much the right ones.  They should all be answered there.   For example, energy is conserved except for momentary fluctuations due to the uncertainty principle. 
There are no separate "messenger particles", maybe the person who used that phrase misspoke.
For electromagnetism the relevant virtual particles are photons, for the strong force "gluons", for the weak force the W' and Z, all of which have been found.  All of these are instances of "gauge bosons".  Again, see Wikipedia for a basic reference.   The inverse square law for EM is related to the zero rest mass of the photon, the short range, and exponential decrease with distance of the strong and weak forces is related to the non zero rest mass of their exchanged virtual particles.  
Gravity is a special case, since there is as yet no quantum theory of gravity.   Stay tuned.
A: The notion of a point-like but charged particle is somewhat contradictory: on one hand we imagine it as localized in space (like a neutral particle); on the other hand, such a "particle" is "long-handed", i.e., it is felt far away from it "position". One may think that the charge is not point-like but is a "part" of a complicated, extended system. When you tear the system apart, you feel a resistance force. In case of a spring, this force increases with distance. In case of a chewing gum, it decreases ;-).
In QED there is a Coulomb gauge where the interaction potential is instant 1/r but acts between "waves", not "points". These waves are "overlapping" and we cannot really separate them.
A: Brief answer: Read only the bold part (and ignore grammar then).
The answer you already mentioned lies in Quantum Field Theory (QFT). But to fully understand it, you must give up a particle as a point-like thing that is well-localized. There is one Quantum Field per sort of particle, e.g. the electron field for all electrons, and the photon field for all photons. (The fact that there is a single field for all electrons also results in the Pauli exclusion principle.)
What you consider a particle is basically just a local peak in the respective particle field, but one cannot even say "This peak corresponds to electron A, this one to B". Now QFT, more specifically Quantum Electrodynamcis (QED), describes the local interaction between the electron field and the photon field. But since the fields have a dynamic, a local change induced in the photon field by the electron field will propagate with the speed of light (flat space assumed) and interact with the electron field in another place, thus creating the impression "Electron A emitted a photon that told electron B to interact electromagnetically".
It's similar for the other interactions, there's a gluon field for the strong interaction (Quantum Chromodynamics), and for the electroweak interaction there's kind of a combination of the photon field and the weak-interaction-bosons.
A: A physical understanding of how forces are mediated is simply a quantum argument.  The momentum operator $p~=~-i\hbar\nabla$ is written in a gauge covariant form $p~\rightarrow~p~+~ieA$, for $A$ the vector potential.  This couples the particle to the electromagnetic field.  The momentum operator is usually expressed in the Dirac equation.  Now if you have two charged particles with momentum $p_1$ and $p_2$ they are coupled to the QED field, even if there are no real photons present.  The QED vacuum couples to the electrons, and this couples the electrons to virtual momenta of the QED field, or a $\delta p~=~A$.  A virtual photon (here thought of on a “tree level”) can then transfer a $\delta p$ of momentum from $p_1$ to $p_2$.  This means the two electrons scatter with momentum $p_1~-~\delta p$ and $p_2~+~\delta p$.  So the presence of uncertainty fluctuations is what transfers momentum from one particle to the other.
A: 
For example, how do two charged particles know that they are to move apart from each other? Do they communicate with each other somehow through some means?

Yes, specifically the electromagnetic field. To give a simplistic view, a charged particle produces an electric field to indicate its presence and a magnetic field to indicate its motion. Any disturbance in these fields propagates outward at the speed of light. Another charged particle some distance away can react to the fields; if it "feels" the presence of an electric field, it will move in response.

I've heard some people tell me that they bounce together messenger photons. So does one electron receive a messenger photon, go, "Oh hey, I should move in the direction opposite of where this came from, due to the data in it", and then move?

The idea of the messenger photon is really just an analogy. Personally I don't think it's a very good one, but we don't really have anything better. The thing is, even though disturbances in the EM field propagate like waves, when a particle reacts to such a disturbance, it acts as though another particle collided with it. In order to make this fit with our intuition, we're forced to invent the idea that there are particles, or quanta, associated with the electromagnetic field, and we call them photons.

Aren't photons also associated with energy, as well? Does this type of mediation imply that electrons give off energy in order to exert force on other electrons?

Yep. To make sense of this, you really have to consider the whole system of both electrons, as well as the EM field itself. Each electron "feels" the electromagnetic field produced by the other, so both electrons' motions will change in such a way that the total energy is conserved.

Every electron is repelled by every other electron in the universe, right? How does it know where to send its force mediators? Does it just know what direction to point it in? Does it simply send it in all directions in a continuum? Does that mean it's always giving off photons/energy?

This is one of those cases in which it makes more sense to think of disturbances in the EM field as waves, which can certainly be radiated in all directions. However, the electron only disturbs the field when its motion (or quantum state, rather) changes. An electron stuck in a single quantum state, such as an atomic orbital, won't be radiating any energy.
The bottom line is that I would advise you not to take the idea of "messenger photons" too literally. It's just a model that works in some situations but not in others.
