It's worth clarifying terminology on what you'll be doing with the barrel adjuster here. You want to screw the barrel adjuster inwards into the derailer body by turning it clockwise. Hence it's "tightening" (in the sense that this is how you turn any normally-threaded screw or bolt in order to tighten it) but it's also loosening (in the sense that you are taking up less cable after the adjustment).

This assumes that by "shifting up" you mean shifting from a smaller rear cog to a larger rear cog (so a gear that is easier to pedal).

Also note that if you have a Shimano "rapid-rise" rear derailer (in which cable pulls the derailer in the reverse direction) you'll need to reverse the directions given in this thread.