Let's start off on common footing by making sure we understand the definition of energy:
Energy is the capacity to do work.
An object may have many types of energy, we'll get to those in a second, but the total energy of an object will be the sum of its kinetic energy and its potential energy. Kinetic energy is the energy associated with the movement of an object and potential energy is the energy associated with an object's position.
To take a physical example, hold out a book (preferably your heavy, expensive general chemistry textbook) and drop it on the floor (preferably, not on your little brother's toes). The book falls to the ground and a loud noise is heard. As the book moved, the potential energy stored in its original position was converted to kinetic energy as it fell to the ground. Upon hitting the ground the individual atoms that make up the book crashed in to the individual atoms that make up the ground, causing them all to move. This movement of atoms is thermal energy.
Let's focus on thermal energy for a moment, the majority of chemical reactions, including the combustion reactions you refer to, involve the transfer of thermal energy. Hold a cup of coffee in your hand; what happens to your hand? Assuming the coffee was hot, and your hand was not, your hand starts to feel warm. We might say your hand is heating up. This thing we refer to as *heat is the transfer of thermal energy; the molecules in the cup of coffee were bouncing around, crashing in to the cup, which made the molecules in the cup crash around hitting your hand.
So now let's turn to chemical energy which is the energy associated with the position of nuclei and electrons in atoms and molecules. Since we are talking about (relative) positions of objects, this energy is a form of potential energy. Some molecules, such as methane, have nuclei and electrons that are positioned in such a way that they store a large amount of potential energy. When the positions of these nuclei and electrons are moved, potential energy is converted into kinetic energy.
So in summary, the energy of chemical reactions is best described in terms of potential and kinetic energy, or in other words the relative position and movement of nuclei and electrons. Often, the energy we experience in chemical reactions can be adequately described by the transfer of heat, and therefore thermochemistry and thermodynamics take up a large portion of our instruction in Chemistry. Other types of energy, such as electromagnetic radiation, electrochemical, and nuclear, can all be described in terms of potential and kinetic energy, namely the position and movement of objects.