Since I'm still banned from the mechanics section (and it's getting to be a bit tiresome) we'll do this here. Do the following, in this order:

1) Shift to the small cog and undo the cable from the derailleur.
2) Confirm the high limit is correct. Pedal the bike and tighten til the chain makes noise, then back it off til it doesn't plus 1/4 - 1/2 a turn. On most drivetrains you want the inside edge of the upper pulley to line up with the outside edge of the small cog.
3) Adjust the barrel adjuster on the rear derailleur all the way in, then back it out a turn or so.
4) Attach the cable, making sure the shifter is in high gear position.
5) Turn the crank and shift once to a lower gear. If you don't get any noise you're ok to continue slowly shifting to easier gears. Shift to the next to the largest cog using the shifter, then use your thumb to carefully push the derailleur the last bit to the big cog. Once it makes that shift stop pedaling and see if you can push the derailleur any further. If you can, adjust the low limit. You should end up w/ the teeth of the upper pulley centered under the teeth of the big cog. Confirm by pushing the derailleur body w/ your thumb to make sure it doesn't go into the spokes. ***If you make the first shift and you get some noise, determine which way you need to adjust by pushing lightly on the derailleur...if the noise stops you know which direction you need to turn the barrel...counter clockwise. If you push the derailleur and the chain immediately goes to the next cog you know you need to turn the barrel clockwise. ***If you get some noise and it seems to get progressively worse as you shift to bigger cogs your derailleur hanger is most likely out of alignment, bent in towards the wheel. EVERY time a bike goes in my stand at the shop with 'shifting problems', especially if it's a kid's bike, I ALWAYS eyeball the hanger from directly behind the bike first. If this is the care, align the hanger. At this point you can check the b gap adjustment. Small ring, big cog. Eyeball the gap between the upper pulley and the large cog. With your derailleur you probably want about 6mm or so clearance. Enough that when you make that last shift to the big cog you don't get any noise/the chain isn't getting pinched between the cog and the pulley. Also, you should confirm your chain is the proper length. Small ring, small cog. Chain should be just short enough it doesn't rub one the pulley cage at all. The cage should also NOT be pulled all the way forward when in the big cog. If it is either your chain is too short OR your derailleur pulley cage isn't long enough. With 1X this isn't usually a problem. If the bike is new I'd compress the housing and make sure the ferrules are seated fully by starting in the small cog, hold the derailleur body in your hand like you're trying to keep it from moving and shift back and forth 1-2 shifts with the lever. Re-adjust as necessary. This will ONLY affect housing length (NOT cable tension...that is not an adjustment, it's always the same) relative to cable length. You NEVER adjust cable tension, you are only changing the length of the housing relative to the cable.
Done.