Front derailleur takes 6 clicks to shift gears, instead of 3 I've just stripped all cables and chain from a badly kept bicycle and replaced them.

I've correctly adjusted the rear derailleur and it works great.

However, front derailleur which only has 3 gears isn't working as expected.

It takes 2-3 steps on the gear to actually change 1 gears.
The range seems to be 1-6, instead of 3.

The gear is the type that is round, circular next to the handles.
This is normal. The procedure is called trimming. 

It allows you to move the front derailleur cage a bit to prevent the derailleur cage rubbing on the chain due to the changing angle of the chain when shifting in the rear.