Shifter not changing gear cable tension
You probably need to check that the shifter is working properly, then set up the derailleur. The problem might be that the cable head is not quite seated in the shifter. 

Disconnect the cable from the derailleur. Find a place to grab the cable where it is exposed - the downtube is usually a good spot. Tension the cable manually and work the shifter, you should feel it moving the cable for all gear shifts. If it's hard to feel the cable moving, stick a 'flag' of tape on it and a marker of tape on the frame so you can see the relative movement.

Once you are convinced the cable is being moved by the shifter properly, check the cable/housing run. Everything needs to be firmly seated in place.

Something else to check at this point is that the derailleur limit screws are adjusted properly and that the derailleur can make it onto all sprockets. Manually push the derailleur while pedaling the crank by hand (be careful not to catch your fingers though). If the derailleur will not go onto all sprockets adjust the limit screws - the Park Tool process should tell you how to do that. 

When you re-attach the cable to the derailleur:


Make sure the shifter is in the highest gear (assuming 'high normal' derailleur) or has let out the maximum amount of cable
Barrel adjusters are screwed all the way (or at least 90%) in
Pull the cable tight to get rid of any slack


You should find the shifter moves the cable nearly immediately as you select the second higher gear. You can then go through the Park Tool derailleur setup process.