Here is how I start up a Cessna 172-S (IO-360 180HP):

Cold start (oil temp below green arc): Master on, Trottle foward,
mixture foward, fuel pump for 3 sec (look for rise in fuel flow
indicator), throttle to 1/4 inch (about 1000 RPM), Ignition

Warm start (Oil temp in the green arc): Master on, Throttle foward,
mixture foward, fuel pump for 1 sec., mixture back to cut-off,
Ignition. When engine fires, smoothly but quickly retard the throttle
to about the 1000 RPM range, and quickly follow with mixture to full
rich.

Hot start (Oil temp in green arc and engine shut down within last 30
minutes): Master on, Throttle foward, mixture foward, mixture back to
cut-off,

Why mixture forward followed by mixture back to cutoff?




When the plane was shut down last, it was accomplished by pulling the
mixture to cutoff and starving the engine of gas. By pushing the mixture
full foward for the hot start, The fuel in the lines primes the engine
without the need for additional priming with the aux. fuel pump. The engine
catches almost as soon as you engage the starter with a hot start.