The exchange of photons gives rise to the electromagnetic force Pardon me for my stubborn classical/semiclassical brain. But I bet I am not the only one finding such description confusing.
If EM force is caused by the exchange of photons, does that mean only when there are photons exchanged shall there be a force? To my knowledge, once charged particles are placed, the electromagnetic force is always there, uninterruptedly. According to such logic, there has to be a stream of infinite photons to build EM force, and there has to be no interval between one "exchange event" to another. A free light source from an EM field? The scenario is really hard to imagine.
For nuclei the scenario becomes even odder. The strong interaction between protons is caused by the exchange of massive pions. It sounds like the protons toss a stream of balls to one another to build an attractive force - and the balls should come from nothing.
Please correct me if I am wrong: the excitations of photons and pions all come from nothing. So there should be EM force and strong force everywhere, no matter what type of particles out there. Say, even electrical neutral, dipole-free particles can build EM force in-between. And I find no reason such exchanges of particles cannot happen in vacuum. 
Hope there will be some decent firmware to refresh my classical brain with newer field language codes. 
 A: 
To my knowledge, once charged particles are placed, the electromagnetic force is always there, uninterruptedly. According to such logic, there has to be a stream of infinite photons to build EM force, and there has to be no interval between one "exchange event" to another.

Nope. Any exchange event requires a finite amount of time and energy. So, no, electrons do not experience a "continuous" force. At the quantum level you would not expect processes to be continuous in general. Also you are conflating two different pictures of interaction. In a classical framework yes the em force is "always there". This is not true in a quantum description. Google checkerboard model to learn about just such a discrete picture of dynamics for elementary particles.
Imagine two people sitting in cars on a frictionless surface throwing snowballs at each other. Every time person A tosses a ball with momentum $\vec{k}$ he/she gains momentum $-\vec{k}$. When person B catches said snowball, assuming a perfectly elastic catch, he/she will gain momentum $\vec{k}$. At the end of this process, A has gained $-\vec{k}$ momentum and B has gained $\vec{k}$. The total momentum of the system is conserved. To an external observer it appears as if A and B have "repelled" each other. How to extend this analogy to induce "attraction" between objects is not clear to me at this time - but let me assume for the time being that it can be done.
Now, for the electromagnetic and gravitational fields, the "snowballs" in question - the photon and graviton - have a zero (or vanishingly small) mass. It doesn't take much effort for A and B to throw these massless snowballs leading to the long-range nature of these forces.

Please correct me if I am wrong: the excitations of photons and pions all come from nothing. So there should be EM force and strong force everywhere, no matter what type of particles out there. Say, even electrical neutral, dipole-free particles can build EM force in-between.

Wrong. Photon and gravitons do not arise from "nothing". To use a cliched phrase - "nothing" comes from "nothing". They are, as @marek mentioned, borrowed from the ground state or vacuum of the field in question. And yes, "electrical neutral, dipole-free particles" can develop attractive interactions purely due to quantum fluctuations. This is the origin of the Van der Waals force.

For nuclei [sic.] the scenario becomes even odder. The strong interaction between protons is caused by the exchange of massive pions. It sounds like the protons toss a stream of balls to one another to build an attractive force - and the balls should come from nothing.

It is precisely because the "snowballs" (pions) are massive, that the strong force is short-ranges and confined to a finite region of space - i.e. the nucleus! Once again, the pions don't come from nothing. Protons, neutrons, pions and all other excitations arise from the same vacuum and should rightly be referred to as "quasiparticles" rather than elementary particles. As to what the nature of this vacuum is, we are still trying to figure that out ;)
A: Update: I went over this answer and clarified some parts. Most importantly I expanded the Forces section to connect better with the question.

I like your reasoning and you actually come to the right conclusions, so congratulations on that! But understanding the relation between forces and particles isn't that simple and in my opinion the best one can do is provide you with the bottom-up description of how one arrives to the notion of force when one starts with particles. So here comes the firmware you wanted. I hope you won't find it too long-winded.
Particle physics
So let's start with particle physics. The building blocks are particles and interactions between them. That's all there is to it. Imagine you have a bunch of particles of various types (massive, massless, scalar, vector, charged, color-charged and so on) and at first you could suppose that all kinds of processes between this particles are allowed (e.g. three photons meeting at a point and creating a gluon and a quark; or sever electrons meeting at a point and creating four electrons a photon and three gravitons). Physics could indeed look like this and it would be an incomprehensible mess if it did.
Fortunately for us, there are few organizing principles that make the particle physics reasonably simple (but not too simple, mind you!). These principles are known as conservation laws. After having done large number of experiments, we became convinced that electric charged is conserved (the number is the same before and after the experiment). We have also found that momentum is conserved. And lots of other things too. This means that processes such as the ones I mentioned before are already ruled out because they violate some if these laws. Only processes that can survive (very strict) conservation requirements are to be considered possible in a theory that could describe our world.
Another important principle is that we want our interactions simple. This one is not of experimental nature but it is appealing and in any case, it's easier to start with simpler interactions and only if that doesn't work trying to introduce more complex ones. Again fortunately for us, it turns out basic interactions are very simple. In a given interaction point there is always just a small number of particles. Namely:

*

*two: particle changing direction

*three:

*

*particle absorbing another particle, e.g. $e^- + \gamma \to e^-$

*or one particle decaying to two other particles $W^- \to e^- + \bar\nu_e$



*four: these ones don't have as nice interpretation as the above ones; but to give an example anyone, one has e.g. two gluons going in and two gluons going out

So one example of such a simple process is electron absorbing a photon. This violates no conservation law and actually turns out to be the building block of a theory of electromagnetism. Also, the fact that there is a nice theory for this interaction is connected to the fact that the charge is conserved (and in general there is a relation between conservation of quantities and the way we build our theories) but this connection is better left for another question.
Back to the forces
So, you are asking yourself what was all that long and boring talk about, aren't you? The main point is: our world (as we currently understand it) is indeed described by all those different species of particles that are omnipresent everywhere and interact by the bizarre interactions allowed by the conservation laws.
So when one wants to understand electromagnetic force all the way down, there is no other way (actually, there is one and I will mention it in the end; but I didn't want to over-complicate the picture) than to imagine huge number of photons flying all around, being absorbed and emitted by charged particles all the time.
So let's illustrate this on your problem of Coulomb interaction between two electrons. The complete contribution to the force between the two electrons consists of all the possible combination of elementary processes. E.g. first electron emits photon, this then flies to the other electron and gets absorbed, or first electron emits photon, this changes to electron-positron pair which quickly recombine into another photon and this then flies to the second electron and gets absorbed. There is huge number of these processes to take into account but actually the simplest ones contribute the most.
But while we're at Coulomb force, I'd like to mention striking difference to the classical case. There the theory tells you that you have an EM field also when one electron is present. But in quantum theory this wouldn't make sense. The electron would need to emit photons (because this is what corresponds to the field) but they would have nowhere to fly to. Besides, electron would be losing energy and so wouldn't be stable. And there are various other reasons while this is not possible.
What I am getting at is that a single electron doesn't produce any EM field until it meets another charged particle! Actually, this should make sense if you think about it for a while. How do you detect there is an electron if nothing else at all is present? The simple answer is: you're out of luck, you won't detect it. You always need some test particles. So the classical picture of an electrostatic EM field of a point particle describes only what would happen if another particle would be inserted in that field.
The above talk is part of the bigger bundle of issues with measurement (and indeed even of the very definition of) the mass, charge and other properties of system in quantum field theory. These issues are resolved by the means of renormalization but let's leave that for another day.
Quantum fields
Well, turns out all of the above talk about particles (although visually appealing and technically very useful) is just an approximation to the more precise picture of there existing just one quantum field for every particle type and the huge number of particles everywhere corresponding just to sharp local peaks of that field. These fields then interact by the means of quite complex interactions that reduce to the usual particle stuff when once look what those peaks are doing when they come close together.
This field view can be quite enlightening for certain topics and quite useless for others. One place where it is actually illuminating is when one is trying to understand to spontaneous appearance of so-called virtual particle-antiparticle pairs. It's not clear where do they appear from as particles. But from the point of view of the field, they are just local excitations. One should imagine quantum field as a sheet that is wiggling around all the time (by the means of inherent quantum wigglage) and from time to time wiggling hugely enough to create a peak that corresponds to the mentioned pair.
A: The original question seemed highly confused IMO.   I wrote a comment to the original question (see above) which I slightly paraphrase as follows:
OK, so we've sharpened your question. You want to know 1. What are virtual particles and 2. How, in principal does their exchange give rise to real forces, e.g. a) how does the exchange of the virtual gauge bosons of EM (virtual photons) give rise to the EM force, b) how do the exchange of virtual gauge bosons of the strong interaction (virtual gluons) give rise to the strong force, and c) similar for W and Z0 and the weak interaction? You do not want the details, but the basic principal of how exchange of virtual particles can yield a real force.
So in answer to question #1 What are virtual particles?
- I suggested he/she start at http://en.wikipedia.org/wiki/Virtual_particle
In answer to questions #2. How, in principal does the exchange of virtual gauge bosons give rise to real forces?  For this, see
See http://en.wikipedia.org/wiki/Force_carrier
In answer to questions #2b and 2c, i.e. give some indication at the undergraduate level how these same sort of Feynman diagrams could also apply to the strong and weak interactions, see 
http://hyperphysics.phy-astr.gsu.edu/hbase/particles/expar.html
For a popular reference book, consistent with the apparent level of knowledge of the original poster, I gave this free reference to The Particle Century by Gordon Fraser for good measure.    IMHO, the original poster is going to be best served by a list of good semi popular books on these issues.  Constructive additional comments to my attempt at an answer might best be served by a list of such books which were thought of highly by the commentator(s), for a) general undergraduates b) physics undergraduates and c) possibly even h.s. students.  
