Well I think each airplane is slightly different.

However I fly most the newer 172SP, and what I have found is that you
need to prime (Fuel pump on, mixture rich) for a little longer than
what the POH specifys. Otherwise it just cranks, and cranks while of
course the fuel that was there to prime it is now gone, so you restart
the process but this time you use even less prime because you think you
have primed plenty the first time. Which makes the issue worse. Also
make sure you have 1/4 throttle, any less and it may be have enough to
get it going. As far as when to advance the mixture.... As soon as it
first starts to catch, advance it in all the way, rather quickly but
smoothly.

One thing you DO NOT want to do (I learned this the easy way, by
watching my instructor do this) is pump the throttle open and close to
force prime it. We (rather he) tried this method and we started
smelling fuel, we got out and saw a puddle of fuel near the front
wheel. We waited 5 mins and it finally started.