What is the best amount of gears on a road racing cycle?

There's no set answer to that, it depends on how fit you are, and how heavy the going is.

Usually, the best gear is the one where you can pedal most efficiently. For a longer ride this usually means whichever gear that lets you keep the cranks turning at a rate of 80-100 turns/minute.

If you bicycle has three chainrings(upby the pedals), then you want to avoid cross chaining - running the chain from the biggest chain wheel to the biggest sprocket, or from the smallets chainring to the smallest sprocket.

It puts an unnescessary bend in the chain, and the gear ratiOS are available at other more chain friendly combinations.