What particles carry various forms of energy? If I didn't get this wrong, light or heat energy consists of photons and they in turn effect electrons' behavior and thus responsible for chemical and electrical energy. What kind of similar particle like photons are kinetic energy of objects?
What about other forms of energy, like potential energy especially in springs (gravity remains tricky I understand)? I'm not specialized in physics so please simplify your answer as much as you can...
 A: Kinetic energy need not be carried by any special carrier particle. Kinetic energy is inherent in any moving body or particle.
If you look closely at the conservation of energy/work energy theorem, all you get is that $$\Delta\left(\frac12mv^2\right)=\int \vec F\cdot d\vec x$$
The left hand side is (change in) "kinetic energy". The right hand side is change is potential energy. Calling both of these terms as energy is a convenience that makes it cleaner to state the conservation of energy in terms of one quantity ($E$).
Fundamentally, kinetic energy is inherent in any moving body. On the other hand, forces are caused by exchange of gauge bosons, so potential energy is in the end "mediated" by various particles (like the photon). I wouldn't go so afar as to say that it is "carried" by particles, though. The forces are mediated by the particles; but the potential energy is still a property of the system.
Springs work on the basis of electromagnetic energy. (All forces like the stress reaction force, forces between materials when you push two together, etc come from electrostatic repulsion between electrons in atoms.)

Photons carrying heat energy and photons mediating the EM force are very, very different.
When you transfer heat, the kinetic (vibrational) energy of the body is manifested in the form of an emitted photon. All that energy is now the kinetic energy of the photon. The photon finally bumps into some other material and is absorbed, turning into electrostatic or kinetic energy. All photons here are real photons.
On the other hand, the electromagnetic force (and the other fundamental forces) is generally mediated by virtual particles (photons). These live on borrowed energy, and don't contribute to the potential energy of an interaction. The energy of the interaction is part of the system, not really tied to any individual entity.
A: All particles gravitate, so they carry gravitational potential energy. (Strictly speaking energy isn't always well-defined in general relativity, but you were asking for an answer low in technical details. Strictly speaking, one should better speak about the stress-energy densitites). They also carry energy associated to all other interactions that they engage in (eg electromagnetic, strong, weak). If they have a mass, that mass has an energy equivalent. So let me see. I guess the only particle that, according to this nomenclature, might get away with "carrying" only one type of energy is the graviton because it neither has a mass nor takes part in any other interaction but gravity. But then, we don't know if gravitons really exist. So leaving out the graviton, I think the answer to your question is: all particles do.
