Is energy only transmitted through electromagnetic and particle radiations? Which are the other ways of transmission if any? If energy does not require any medium for transmission(as for sunlight reaching earth, the heat too), is it transmitted in quanta in particle radiation too?
 A: Energy transfer can be thought to occur via the exchange of a 'virtual particle'.  In nature, there are 4 fundamental forces, namely:
1. Electromagnetic force
2. Gravitational force
3. Strong force
4. Weak force
Each of these forces have a different exchange particle: 
For instance, the exchange particle for EM is a photon whereas that for the strong force is the gluon.  The nature of the interaction is characterised by the properties of the exchange particle.
Now if you want to connect this rather abstract idea to a bigger picture of the more 'real world', you just have to carefully think about what the process you are considering actually involved on a deeper level.
For example: suppose you are pushing a box across your room.  What you are actually doing is repelling the electrons on the box by the electrons on your hands, thus causing it to move.  Therefore, you the interaction is an electromagnetic interaction and hence the exchange particles involved are photons.
If you think of energy transfer in this way, then indeed all energy transfers occur via 'particle exchanges' or radiation (since you a particle is essentially a wave packet [wave particle duality]).
A: To expand on the above answer by @PhysLab (which is very well written), the answer depends on how "in depth" you want to look at it. Before discussing energy transfer, let's cover a few fundamentals, because these are slippery concepts and often misunderstood. (Including by me, so fingers crossed!)
Let's look at what "energy" and its "transmission" involves. In an everyday sense, we have two main ways of thinking about it. 
We can imagine that there is energy in one place or one form. It then moves, or changes its form. Examples - The chemical energy in gunpowder changes to energy of heat and expanding gas (chemical reaction) when it explodes. A photon hits a cell in your eye causing a chemical/electrical change that your nerves can transmit to your brain and which you can interpret as "sight". You eat food (concentrated energy) and this powers your body leaving behind waste material.
We can also imagine energy as being involved in letting or making something happen ("doing work" as physicists call it). When a ball rolls down a hill, or you release a taut spring, the energy is kind of initially "hidden" - it's the energy that the ball has because of its raised position (in the earth's gravitational field) and the energy that the spring had because someone/something stretched it first. (We call this "potential" energy if you've come across that term.)
But if you look closer, this is all a bit like the nice solid image on your TV that is really made up of dots. What looks "obvious" isn't really what's going on. The gunpowder is a mix of atoms and molecules which can rearrange themselves in a more stable configuration. They needed more energy to hold themselves in the original unstable configuration ("binding energy" in the form of chemical bonds made up of interactions between electrons in the atoms). The explosion is that extra energy, no longer needed to "bind" the gunpowder in its original configuration, being rapidly given off as heat and energy of motion ("kinetic energy") once there is a route by which the gunpowder can reach that configuration(the initial spark or whatever). The photon interacted with an electron in some cell in your eye, and the result is an electron which has a momentary higher energy level, and when this extra energy is released again, this electrical/chemical change is what is detected. The ball and spring had energy because of their situation, and in the spring's case because something else had preciously pulled its particles slightly more distant from each other, and when that ceased to be the case the energy it had used to do so was released - the ball could find a more stable position (in energy terms)  by moving towards the earth and the springs particles could do the same by moving closer to each other again.
But what you are asking is more like, what fundamentally goes on.  For that, we need to rethink what "energy" actually is. At the best understanding we have at the moment, particle physicists discuss "fields" which follow mathematical rules. So we can talk about an "electromagnetic field", or a "gravitational field". It seems to follow certain rules and behave certain ways. Some particles (and therefore some waves as well, because particles and waves are the "same thing" at this level) interact with these fields and therefore these particles can be thought of as "noticing" their presence. 
Electricity is a good example to explain this better. Electricity as we know it, is the result of an electrical field. It arises from the electroweak field. But if nothing responded to it or interacted with it, it wouldn't actually exert any force and we probably couldn't detect it. As it happens, some particles do respond to it (electrons, protons), and some don't (neutrons). So an electromagnetic field will interact with electrons and protons, but not with neutrons. As far as the neutron is concerned the electromagnetic field isn't there. But the interaction between the electromagnetic field and those particles that do interact ("couple") with it, means that at every interaction, energy can be transferred/redistributed between the field and anything else involved or resulting, and our use of electricity is pretty much just exploiting that fact and what we've learned about the rules that govern it. 
Gravity is another field (the "gravitational field") and interestingly, all particles seem to interact with it. 
The Higgs field (of "Higgs Boson" fame in the news a while back) is a field which some particles interact with and some don't. Just like particles that interact with an electric field behave as if they have a property called "electrical charge" and those that don't interact with it behave as if they have no electrical charge, so particles that interact with the Higgs field behave as if they have a property we call "mass" while those that don't (like photons) behave as if they are "massless".
Coming back to the original question, what this all does is turn the question you asked on its head.  So far as we know, all energy is transmitted when fundamental field(s) interact either between themselves or with those particles able to interact with them. The exact circumstances of that interaction determine what happens, and the outcome might be that energy is now in a form it wasn't before (a new particle "created", some photons, a higher or lower energy state or configuration which then itself changes, and so on). That outcome, of a changed form of energy is ultimately responsible for all the interactions we know of, all transmission  and movement of energy we know of, everything from fusion to falling objects to elastic to sight to chemistry to lasers to how proteins fold and how enzymes work .....
