A friend I took on a local flight recently recorded a video of the engine start-up sequence for the Cessna 172SP. It's the regular start-up sequence straight from the flight manual. The engine is a fuel-injected Lycoming IO-360-L2A.
Since this was the first flight of the day, the engine had to be primed prior to start-up. This is done by setting the mixture to full rich (pushing the red knob all the way forward) and turning the electric fuel pump on for about 6 seconds, until the needle on the fuel flow gauge comes to life and gives a stable reading.
As surprising as it may sound, a stable needle on the fuel flow gauge does not indicate a stabilized fuel flow. It indicates a stabilized pressure in the fuel lines.
The distinction between pressure and fuel flow does not matter in flight: the position of the needle on the gauge is driven by a fuel pressure sensor and the gauge is marked in gallons per hour because fuel pressure and fuel flow always go hand in hand. Someone in the Lycoming lab probably one day used a piece of equipment that directly measures fuel flow and painstakingly painted ticks on the face of a fuel pressure gauge. The final touch consisted in writing FUEL FLOW in big vertical capital letters on the right-hand side of the needle.
Things are a little different at start-up because, of course, the engine is not yet running. Fuel lines most likely contain fuel vapors and not liquid fuel because liquid fuel evaporated or dripped out since the engine was last shut down.
The role of the electric fuel pump is to push fuel down the fuel lines to the fuel manifold valve and from there to the fuel discharge nozzles, effectively filling up the lines with liquid fuel and getting rid of fuel vapors. Because fuel can only escape through the tiny holes of the discharge nozzles, pressure in the fuel lines rises and stabilizes. That's when the needle on the fuel flow gauge stops moving. The engine is ready for ignition.
Ignition needs to happen as soon as possible after priming, otherwise fuel may drip down the air intake pipe and cause a fire hazard. This is because this engine features indirect injection which delivers fuel at the intake port, i.e. upstream from the intake valve that allows the fuel/air mixture into the cylinder. The other sort of fuel injection, direct injection, delivers fuel directly into the combustion chamber of the cylinder, and is referred to as "common rail" in car marketing brochures.
Turning the ignition key makes the electric starter motor rotate the engine, which makes the magnetos deliver high-voltage electricity to the spark plugs with some help from the impulse coupling, which in turn ignites the fuel-air mixture in the cylinders.
One peculiarity of starting this fuel-injected engine is that it is done with the mixture on idle cut-off, i.e. full lean. The mixture is then quickly advanced to full rich once the cylinders start to fire. Why this is so I do not know. According to a thread I read on pprune.org some time ago, fuel-injected Continental engines work the opposite, they are started with the mixture in the full rich position. Go figure. I guess that's the point where pilots should stop asking questions and just trust what the POH says.