The barrel adjuster adjusts the position of the derailleur with respect to the cassette. If you are shifting from one cog to the next and it is taking more than 1/2 pedal revolution to compete the shift then you can adjust with the barrel adjuster. Same if you shift from one gear and it skips and goes 2 up instead of just one.
It does do this by adjusting the tension but does not sound like it is the cure for your problem. I am not sure what you mean by hard shift. The lever is hard to move? If that is the case then it could be the tension is to much to begin with. It is always easier to start from scratch and set up the derailleur through the steps from somewhere like Park tool or Sheldon Brown than to try to guess at adjustments without knowing the effect.
For adjusting the front derailler small ring, the rear should be on the largest cog in the back. The derailleur low limit setting should be just enough to allow clearance. When adjusting the FD large ring, rear derailleur should be on the smallest cog in rear and high limit should be set to just allow clearance.
Kenal0