Combustion is a very interesting topic in science because it ties together a remarkably large number of facets of science into one neat little package. Oh, and because its fire! Of course it's going to be interesting! There's all sorts of little nuances that go into how fire works. It's worth going over them all, to give you more opportunities to weave your magic in.
Fundamental to combustion is the idea of a reduction-oxidization reaction, typically known as "redox." In a redox equation, we combine two materials, a reducing agent and an oxidizing reagent and they interact to form new compounds. In combustion, the reducing agent is our fuel and the oxidizer is... well... oxygen in the air! However, it may be helpful to start with a simpler redox equation: the creation of sodium chloride.
$$2Na+Cl_2\to2NaCl$$
This is the chemical equation governing the conversion of sodium and chlorine into table salt. Yes, you can actually make table salt out of a highly reactive metal and a poisonous gas. Now in chemistry, one of the most important properties of any atom is how many "valance electrons" it has. Atoms pack their electrons into different orbitals, each with more energy than the last. The low energy ones can't interact with other atoms much, so we ignore them, but the ones in the highest energy orbital do interact. This is such an important feature that we organize our periodic table around it!
Please ignore the metals highlighted in green for now. The same rules will apply, but they're more complicated. Rest assured, you can burn them.
So let's look at our example of sodium (Na) and chlorine (Cl). From the table above, sodium has 1 valence electron and chlorine has 7. Now as it turns out, atoms are most stable if they have 8 valence electrons or 0. Why? We can model the reason for it in Quantum Mechanics, but I think that's digging too deep, so we'll just take it on faith. Neither sodium or chlorine have 0 or 8.
However, they can share.
Sodium can't just give away its electron, because then it would be charged and that's actually less stable. However, it can let chlorine borrow it. Think of it like loaning someone a pen. You can loan it to them, and they can use it, but you never let them out of your sight. If you do, they'll walk off with your pen and you'll never see it again. So in the case of sodium and chlorine, sodium loans an electron to the chlorine so that sodium has 0 valence electrons, chlorine has 8, and everyone's happy. They just have to stick together now, so that sodium doesn't lose track of its electron like you can lose track of a pen.
In this reaction, sodium is known as the "reducing agent," and chlorine is known as the "oxidizing agent." Why are we calling something other than oxygen an oxidizer? History. The earliest redox equations explored used oxygen as the oxidizing agent, so the name made sense. Later, it was expanded to include other compounds besides oxygen which could do the same sort of thing. Chemistry has a whole tool dedicated to undersatnding this, called Oxidization States, but that can be explored later.
Okay, first chemistry lesson over. Now let's talk about fire. When we have combustion, the fuel is the reducing agent, and oxygen is the oxidizing agent. This is a definition -- if you don't have oxygen reducing a fuel, it is simply no longer called combustion. There are many such reactions. Here's two examples:
$$Mg + O \to MgO$$
$$C_6H_{12}O_6 + 6O_2 \to 6CO_2 + 6H_2O$$
The first reaction is the burning of magnesium. This is a highly energetic reaction used in survival kits as a fire starting mechanism. The second is the burning of glucose -- if you threw a peep onto a fire, this is similar to the reaction you would see (it's also the reaction your body is using to burn glucose for energy!). The key to both of these is that, if you look at how many valence electrons each atom has, they have begged borrowed and stole to get closer to either 0 or 8.
Redox equations are almost always exothermic. That means they generate heat. The reagents had energy bound up in how they deal with those unsightly 1 or 7 valence electrons, and once they start sharing that energy gets released. How much heat can be calculated from enthalpy equations, but there are plenty of tables of how much energy is generated by burning each material, such as wikipedia's Energy Density table. From that table we can see that burning sugars (like above) generates 17MJ/kg, and burning gasoline produces 46MJ/kg.
However, this must be only part of the story. So far, everything I have talked about makes it look like everything is constantly combusting. There's oxygen in the air, so why isn't my desk catching fire? The answer is that these reactions are indeed happening all the time, they just happen slowly. To discuss this, we need to move to the second layer of the chemistry: kinematics.
In chemistry, kinematics is the study of how fast a reaction occurs. In general, it takes energy to cause a reaction to occur. This makes sense, in that if it didn't take energy, you'd never see the original reagents in nature -- they'd always have found a way to combine until they were at the lowest energy possible. Chemists call this "activation energy." In the reaction of sodium and chlorine, the activation energy is the energy it takes to push the atoms close enough together to start sharing electrons. Most reactions have an energy curve associated with them that looks like this black line:
The timeline starts on the left, with the reagents in their natural state. We apply some extra energy to raise them up to the transition state, and then they release all of that energy and more as they combine to form the final products. Most often this energy comes in the form of kinetic energy. Atoms are constantly in motion with some statistical distribution of velocities. At all times, some of the oxygen molecules in the air are moving fast enough to bring enough energy to the reaction to cause a redox equation with the wood of my desk, but most of them are moving too slow.
This is where we add heat. If we raise the temperature, more molecules are moving fast. This means more of them are moving fast enough to exceed that activation energy and enter a redox equation. Now we're getting close to something we would call fire. Each redox reaction generates more heat, which means it creates a bunch of byproducts that are moving fast (like fast moving $CO_2$). As these flee the scene of the crime, they may bump into oxygen atoms and transfer their energy to the oxygen, or they may bump into the nearby wood molecules. If they transfer enough, they'll have raised the temperature of the area to make it likely that the next oxygen atom can cause combustion too.
If this process generates enough localized heat, it creates a runaway reaction that we call fire!
So with all of this chemistry, there's countless options for your characters. I'm assuming that they don't get to violate the conservation of energy, so the static part of redox is not going to give you much room, but the kinematic side is loaded with opportunities. The obvious approach is to let them raise the temperature of the air until something ignites, but the kinematics show a few other options. In normal situations, the speed of molecules in the air is described by a Maxwell-Boltzman distribution. This is a very simple distribution that occurs in nature, where high energy particles are very rare. If your characters could change that distribution to a bimodal distribution, with more fast particles and a bunch of really really slow particles, they could cause combustion to start faster. This is actually what your refrigerator does (though in that case we are trying to use the slow particles, not the fast ones). However, this approach will only work to start the fire. Once it's lit, it will have to sustain itself, because they can't just spend all their energy creating funny distributions.
You can also try to rarefy the air, moving oxygen closer to the fuel. If the air near the fuel is more oxygen rich, it is more likely to combust. This is why hospitals have so many safeguards when dealing with oxygen tanks. In the presence of extra oxygen, some things will combust that wouldn't have before, simply because the kinematics permit the runaway reaction sooner. (In fact, in SCUBA diving, there's something known as oxygen toxicity. It occurs when the partial pressure of oxygen gets high enough to start corroding your lungs!)
The most common solution, however, is a catalyst. Catalysts simply reduce the activation energy of the reaction by letting it occur using less energetic transient states. If you decrease the activation energy of the reaction, a larger portion of the molecules will be lucky enough to have enough velocity to react. This is what we're doing with the catalytic converter in a car's exhaust system. The conversion of the nitrous oxide compounds to nitrogen and oxygen is natural, but slow. The catalysts in the catalytic converter reduces the activation energy of this reaction, so that it happens fast enough to complete before the exhaust leaves the car.
The best example of a catalyst in the world is catalase. Catalase is an enzyme produced in almost all living creatures to break down hydrogen peroxide. Hydrogen peroxide in the blood is a terribly powerful reducing agent which causes lots of problems, so we don't like it. It wants to break down into water and oxygen naturally (decomposition, not a redox reaction), but that process is slow, taking hours or days. Hydrogen perodixde usually causes its damage by then. The reaction is slow because the activation energy for this is quite high. Catalase decreases that activation energy so serverely that instead of taking hours it takes nanoseconds. We actually have trouble measuring how fast the reaction goes in the presence of catalase because its too fast! You've seen this reaction if you've poured hydrogen peroxide on a wound and watched it bubble. The bubbling is oxygen released from the decomposition catalysed by catalase.
The neat thing about catalysts is that, unlike the other methods, it can also help keep a fire going. Catalysts are not consumed in a reaction, so they can continue to affect it until the temperatures eventually break the catalyst down.
So hopefully those help. Catalysts are probably the way to go, but it's helpful to understand what fire actually is, so that you can explore all of your options.
