Where do photons go when they are absorbed?

Well, the answer you usually get is half right. They do disappear (more on this in a second). I'd hesitate to say they turn into "heat energy," both because we don't use the term "heat" that way in a technical sense and because most of the time we like to talk about atoms absorbing photons. In this case the energy of the photon becomes potential energy of the electron that made the transition, and there's no need to talk about heat.

Now, can the photon disappear? The short answer is yes. When you talk about things "not simply poofing out of existence" what you're really describing is like a conservation law. For instance, we say that energy is neither created nor destroyed. Your intuition that things aren't just "poofed" out of existence is probably due to your everyday experience that objects generally can be broken into parts, but not usually destroyed. This isn't true in the particle physics sense, usually. The energy carried by that photon has to be accounted for, as does its momentum and angular momentum. But "photon number" is not a conserved quantity the way that energy or (for instance) electric charge are. A photon really is just a way of looking at disturbances/excitations in the electric field, and so its "destruction" just represents that energy that was present in the field has been moved into some other mode.

When you turn on a lightbulb, you easily create many photons. They can go away just as easily. That's because they are bosons and they have no charge.

Think of waves on a pond. Where do they "come from" when you throw a stone in? Where do they go when they dissipate?

That's actually a very good analogy in some ways because the math that describes transverse waves is the same, but different in a very fundamental way: the waves are quantized.

In quantum field theory, the field (the pond surface) is everywhere, and it may become excited (throwing a stone in). It's the additional step of "all or nothing" that gives you particles, but that's another step added on top of the issue. Where does a lump in a hall carpet go if you manage to stomp it out rather then shift it? The lump is not a "thing" but a "state". (I'm reminded of "where does your lap go when you stand up?". It is funny in ascribing thing-ness in the same way as an object, but it is a description of a state, not an atom of matter.)

How are photons created?

An accelerating charged particle generates photons tangentially as well as a decelarating one. Where do these photons come from? From the energy carried by the electron. In this sense photons are just a packet of energy which is associated with the electromagnetic field. This type of interactions of electrons and ions with fields happens in the photsphere of the sun, for example, generating the light spectrum we observe.

A photon can interact with charged particles and give up part of its energy or even all of it, and then it "disappears".

Photons can also be produced when electrons that are bound in atoms by the electric field of the nucleus , in steady orbitals but in an excited energy level, fall to the lower energy level releasing a photon. A photon of the same energy will be able to kick the electron to the higher energy level, disappearing in the process.

This appearing and disappearing is not an attribute of photons only. In general particles meeting their antiparticles disappear, because all quantum numbers add up to zero. An electron meeting a positron disappears into two photons. Where do the electron and positrons go? The photon is a simpler particle as it has less quantum numbers to conserve, but the phenomenon exists for all particles in special situations, which you will find out if you carry on to study physics.