I understand that fire heats its surroundings via conduction, convection and radiation. I've read that conduction is nearly irrelevant to this process as air is a poor heat conductor. In descriptions of convection, people often just say "fire heats the air and the air circulates heat to the environment". But, if air is a poor heat conductor and a poor absorber of radiation, how is the air heated in order to convect heat in the first place?
Fire is a reaction that includes the air. To be precise, it includes a part of the air, oxygen. It releases the results of the reaction as a new part of the air. Those reaction results are hot and mix in with the rest of the surrounding air.
Air is a poor conductor when there is no ability to mix. The "output" of fire (for our purposes assume it is carbon dioxide) mixes with the air, raising the average temperature.
Heat conductivity is not the same as the ability to absorb heat, only to pass the absorbed heat through the material. Air within the flame can absorb heat, but the air cannot quickly pass the heat to neighboring air molecules. This is partially due to the average distance between neighboring molecules within the air. With convection, the high energy molecules move out of the way and low energy molecules move in to gain energy from the fire.
This is why a good window has two parallel panes of glass. Outside air moves around easily, so outside air molecules quickly change the temperature of the outer pane. Molecules effectively contact the glass and a small amount of energy transfers. New molecules contact the outer pane and more energy transfers. Energy can also transfer between the outer pane and air molecules that contact the inner surface of the glass. These air molecules between the panes don't circulate much, so the energy cannot pass quickly from one pane of glass to the other.
That is the reason fire is hot.
While others have pointed out the role of the exhaust products, the main takeaway is that yes, air is not easily heated by fire.
What happens is that the excess energy from the fire piles up and the temperature rises to hundreds of degrees. Both conductance and radiation scale with temperature, so this continues until an equilibrium is reached, when the temperature gradient gets steep enough to compensate for the poor conductivity and the radiation (mostly through the air) together balance the produced heat.
Fire thus eventually start to heat air when the temperature difference gets high enough to compensate for the insulation nature of the surrounding medium.
The fire is a sustained combustion process resulting from the combination of oxygen from the air, fuel and heat sustaining an exothermic chemical reaction that produces various gases (CO2, CO, etc.) and particulate matter at elevated temperatures. They, in turn, rise and mix with the air above the flames creating a mixture at elevated temperature.
But the primary mechanism for heating air not directly above the fire is indirectly by radiation. For example, radiant heat from a fire in a fireplace in a room first elevates the temperature of the various solid materials in the room. Then the temperature of the air in contact with those surfaces increases by natural convection over those surfaces.
Hope this helps.
The air is part of the chemical reaction which produces the "heat" and the gaseous products of combustion have more kinetic energy than the air had originally.
The products of combustion transfer kinetic energy to the local air molecules via collisions.
The movement of the air (convection) helps with the mixing of the hotter gaseous products of combustion and colder air.
Fire generates heat in several ways: mostly as radiation, but also as the kinetic energy of the products of the chemical reactions (which are mostly the products of oxidation of hydrocarbons and carbohydrates: $H_2O$, $CO$, $CO_2$, etc. - see Can we call rusting of iron a combustion reaction?) The air immediately around the fire is thus heated via the energy exchange when these molecules collide with the colder air molecules. The diffusion path of molecules in air is actually rather small - about 10cm (see, e.g, discussions in thread How does hot air rise?.)
On larger scale the air propagates via convection - this is how, e.g., it rises over the chimney or how the upper floor of the house is heated when there is the fireplace lit on the lower floor (eventually, rising of air is also driven by the fact that the length of molecular collisions is short, see the previously quoted post, but also What is (local) pressure within a gas on the microscopic level?.)
But in the latter case we are dealing with the hot air in the room, which is not heated via direct contact with the fire, but via its contact with the objects and the walls, heated by the fire radiation (mostly in the infrared range, see Does fire emit black-body radiation?, If air is a bad conductor, how does fire heat up a room?)