How do you properly adjust a Shimano derailleur on a bicycle?

To properly adjust a Shimano derailleur on a bicycle, start by shifting the chain to the smallest rear cog and the smallest front chainring. Then, adjust the high limit screw to align the derailleur with the smallest cog. Next, shift to the largest rear cog and largest front chainring, and adjust the low limit screw to align the derailleur with the largest cog. Finally, fine-tune the indexing by adjusting the barrel adjuster to ensure smooth shifting between gears.