FBinNY 

First of all, make sure you have 8 speed shifters on an 8s cassette. Count clicks an 8s will have 7 clicks (start + 7 changes = 8). If that's the case your problem is likely one of two common issues.

The hanger may not be aligned, so the derailleurs idler cage isn't parallel to the wheel. You can spot severe misalignment by standing behind the bike, and seeing if the cage is vertical (when the wheel is). However, slight misalignment is hard to see and requires a tool to measure and correct.

The other common issue is cable friction, especially in the rear loop, or on older bikes possibly in the long HB to frame section. Try this -- shift to the 2nd or 3rd low gear sprocket, and adjust the trim so it's perfectly centered. If all is right you should be able to shift to high (maybe not smoothly yet) and shift back to the lower I(not lowest) gear. Adjust so the downshifts are spot on and it trims well in these gears.

Now upshift one position, if it hangs or doesn't trim well, pluck (hard) the bare wire away from the down tube like a guitar string. If this completes the shift and it's now in trim, you know it's a cable friction problem.

Start by testing the rear loop, remove the rear wheel, and move the RD by pulling the wire away from the down tube. It should move smoothly and precisely in both directions. If it's jumping, or hanging when you ease the wire back in, you have friction in the rear loop. OTOH if all is smooth as silk, then the problem is in the long HB section.