While many new airplanes, including trainers, do have fuel injection systems that can make starting a cold engine a little easier, there are plenty of venerable and reliable carbureted aircraft in the skies. At the heart of the difference between these birds and your basic car is the fact that you don't just hop into an airplane and turn the key.
Aircraft carburetors are mounted below the engine, so working the throttle lever back and forth does little more than fill the airbox below with raw fuel. (And that assumes that you're flying an airplane whose carburetor is fitted with an accelerator pump; many trainers are not so equipped.)
Admitting an extra slug of fuel to the cylinders - in a place and quantity that will do some good for getting the thing started - is the job of the primer. Plumbed directly from the fuel system, downstream from the fuel tank selector or shutoff valve, the primer system completely by-passes the carburetor. When you pull on the primer's plunger in the cockpit, fuel is drawn from the lines, usually just beyond the gascolator, and then pumped straight to the intake ports of the engine, just upstream of the inlet valves.
Sending the raw fuel to the intake ports helps to ensure that when the engine is first turned over a sufficiently rich fuel-air mixture will reach the combustion chambers. Cold engines need far richer mixtures to initiate combustion than is required for continued running once the pistons and cylinders begin to warm up. That means the colder the ambient temperature - and hence the engine components themselves - the more you will need to prime.
How much? Follow the recommendations in the pilot's operating handbook if you're unfamiliar with a specific aircraft's requirements. Then have a look in the amplified procedures section if the cryptic note "Prime - As Required" doesn't seem like much help. Generally start with the least amount of priming set out in the guidelines. The worst that will happen if you under-prime is that you will get additional practice starting the engine; an insufficiently rich mixture will allow the engine to start briefly, then die. Over-priming, however, can send enough fuel to the cylinders to wash the oil film from them - a bad thing for piston-ring and cylinder longevity.
If you are operating in very cold climes, leave the primer knob unlocked when you begin cranking the engine. This allows you to pump the knob and admit extra fuel directly to the combustion chambers, which will help keep the propeller turning while the powerplant comes up to operating temperature. Atomized fuel droplets from the carburetor have to travel a great distance and against gravity to gain entry to the cylinders. At low temperatures, this fuel will fall out of suspension from the surrounding air very easily, resulting in a nearly incombustible mixture. Also, during cold weather, pulling on carburetor heat after the engine starts will help this atomization. Use caution taxiing with carb heat on because the intake air will be unfiltered and you could get dust and other debris in the carburetor.
Should you prime when the engine already is warm? It depends. In warm weather, no; the engine will usually retain enough heat to start easily without priming for more than an hour after shutdown. When it's cold, though, you may have to prime the engine even though it hasn't been shut down for long. If in doubt, try to start without priming; if the engine fails to catch within a few revolutions, stop and then prime using the minimum recommendations.
A final technique note - make certain that you pull the plunger out slowly enough for fuel to enter completely; you can usually hear the primer filling. And don't forget to latch the primer closed before takeoff. If left partly extended, fuel will siphon from the primer into the intake tract, making the overall mixture overly rich and robbing the airplane of takeoff and climb performance.