How to lean the mixture properly depends on a number of factors. Air density, quality of the induction system, carbureted or injected, turbocharged, -normalized or normally aspirated, amount of instrumentation you have, fixed pitch prop or constant speed, ...
The engine manufacturer has taken all these into account, and the airframe manufacturer has lifted the bits from the engine manual that apply to that particular configuration, and put it in the POH. So your first and best source of advice should be the POH.
From your post it looks like you are flying a normally aspirated, carbureted engine, and most likely with a fixed pitch prop and limited instrumentation. In other words: A basic trainer such as the PA28 or C172. In those situations, the advice in the POH will basically come down to this:
1. When using full power below 3000 or 5000 feet (the number varies slightly), use full rich.
2. When using full power above said number, lean the mixture for maximum RPM.
3. When flying with any power setting lower than 75% (cruise and descent), regardless of altitude, lean until the engine runs rough, then enrich to restore smooth running.
Enrich the mixture before making any power change, lean as per the above when you are again in a stable configuration - climb, cruise or descent. So even during a cruise descent you can lean until rough running. And in preparation of a possible go-around, enrich the mixture somewhere as part of your pre-landing action.
If your engine configuration gets better (injected vs. carbureted for instance) and if your instrumentation gets better (EGT, per-cylinder EGT, CGT, per-cylinder CGT, ...) you will find the POH prescribes more refined methods. But the crude method above will already be very close to optimal.
Oh, and also note that a properly leaned engine will run with hotter EGTs than a full rich engine. That makes carb heat - which draws air from a muff around the exhaust - more effective.