yaay thermal class to the rescue.
Internal energy (U) is a combination of several things. In thermodynamics, there are a few "fundamental relations" which are said to contain all the information about a system. From these you can do some math tricks to learn basically anything you want, given a few easily-measurable values.
The one for internal energy is "dU = T*dS - P*dV + μ*dN". If you know calculus, then those "dx" variables are actually differentials. It's a multi-variable differential, though, since U is actually a function of (S,V,N). It's analogous to the 1d case you're probably used to, though (dy = dy/dx * dx) If you don't know calc, you can think of them as "a really small change in." The variables are defined as: T = temperature, S = entropy, P = pressure, V = volume, μ is the chemical potential of whatever particles are in your system, and N = number of particles. If you have multiple types of particles, you just add those onto the end (μ1*dN1 + μ2*dN2 + ...).
You can change the internal energy of a system by varying any of those parameters.
What Nicolas said (dU = δQ − δW) is also true(ish). This is because δQ = TdS, and δW = PdV. He left off the μdN part, because in most cases the number of particles in a system is considered constant. This means dN = 0, so the third term in the equation I gave is often left off. I should also note, there's an ambiguity in the way "Work" is defined that will often switch the sign on the P*dV term. We defined dW = -P*dV, but used a slightly different concept for work than what Nicholas' teachers probably taught.
Heat itself is not, strictly speaking, "energy". Like several people have already said, "heat" really isn't a "thing" at all. It's just the name physicists give when that certain type of internal energy is transferred (the T*dS part of the equation).
Judging by all this, I don't think a rolling boulder would be said to have heat energy simply because it has kinetic energy. Entropy has to do with the logarithm of the probability of the molecular state of a system. A rolling boulder, on a macroscopic scale, isn't really any more "likely" to be rolling than still... so the change in entropy from a stable-boulder to a rolling-one is 0, meaning that no heat energy was transferred into it.
^^I'd appreciate it if the other people who know physics well read that over and confirmed or argued me on it, though. I know what the equations say, but while I was typing it I realized a few paradoxes that I'm gonna have to ask my teacher about, if it was all correct (namely, if you have two compartments in a box, with a wall in the middle, and punch a hole in the wall, the entropy of the system will rise as the molecules spread out. Does this mean that, since its T*dS is positive, it gains internal energy simply by spreading, or does the increase in volume (from the -P*dV part) negate it?)
Let's see... AI was right in that the sun is sustained by nuclear reactions. When two hydrogen atoms fuse together to form helium, the mass of the resulting helium is less than the sum of the two hydrogen atoms that created it. Where did the mass go? Into Energy, via E=m*c^2. Since c=3*10^8, c^2 = 9*10^16... meaning that you can get huge amounts of energy out of a very small change in mass. This is why nuclear bombs are so incredibly powerful.
I couldn't really think of any other kinds of energy outside of that link Anthile posted. If you want a more general way of looking at it, energy can be found anywhere where forces are/were necessary to put the system in the state it is. When you have to exert a force to lift up an object in a gravitational field, you're adding potential energy to the system. When you push an object to move it, you're adding kinetic energy to it. When molecules from a hot object bump into molecules of a cold one, they push them, adding thermal/heat energy. When you separate two atoms in a molecule by pulling hard enough on them to overcome the electrons holding them together, you pull chemical potential energy out of the system. And so on for the other ones.
The one notable exception is in magnetic fields--magnetic fields exert forces on charged, moving particles, but do not add or remove energy from them. They're an odd-force-out because the direction of the force is actually perpendicular to the direction of the particle's motion. If you've had something called a "dot product" yet, it's because energy is added or removed only when (force vector) (dot) (direction of motion) is non-zero. If you haven't had dot-products/vectors, then you can take this to mean "if the force is exerted on the particle in a direction that is anything-but-perpendicular to its motion." That's virtually all forces... but like I said, magnetic fields are an odd-one-out.
For the last question: some types of energy can be transferred directly from one form to another, but others can't (I don't think). My lifting an object, giving it gravitational-potential-energy, will never make it heat up. Dropping it, transferring from potential to kinetic, will never make it emit light. There are usually clever little ways to turn most types of energy into other types, though. TVs and screens translate electricity into photons, power generators turn nuclear/heat into kinetic, and then into electricity. Batteries turn chemical potential into electric. Etc. As a general rule of thumb, though, if "energy type A" has to go through "some sort of form B" in order to get to "energy type C", then "form B" is just another type of energy too.
