Picture the barrel adjuster as a simple screw that threads into something (derailleur, shifter, etc.). So for the adjuster on the rear derailleur, look at it from the rear of the bike. Clockwise rotation will decrease cable tension, counter-clockwise rotation increases it.
In your case, it sounds like your cable tension is set too high. Turn the barrel adjuster clockwise 1/2 turn or so at a time and see if that helps. Cable tension should be as tight as possible while still allowing smooth upshifts (to smaller cogs).
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In memory of Jim Price (aka. sydney) ...