New bike rear derailleur not working correctly
Here's a way to set an initial cable position for your rear shifter.

First, if required, open the limit screws on the rear mech so that the chain can get out to the biggest and smallest cogs.  If you didn't change these, they should be okay.


Set the right-hand shifter on the hardest setting (that's the one with the most slack on the wire)
Set the chain on the second-smallest cog.
Set the barrel adjuster to between 1/3 and half-way in.
Relax the pinch bolt on the rear mech that is holding the inner cable. 
Make sure all the cable outers are in their stops, both in the frame and up at the shifter.  They can pop out sometimes.
Pull through as much cable as you can without significantly moving the rear mech, and tighten up the pinch bolt enough for testing.


Now as you hand-pedal the cranks, the chain should drop into the smallest cog.  Work through the gears while slowly pedalling, and the chain should move one cog per click.

If you press the shifter for an easier gear (larger cog) and the chain isn't quite moving up, then back out the barrel adjuster a half-turn at a time.

Likewise if you pull the release for a harder gear and the chain doesn't move down, the wind the barrel adjuster in a half turn.  Check through all gears whenever you make an adjustment, because everything effects everything.

Two last tasks, 


make sure the pinch bolt is done up tightly enough to not slip but not so tight it strips the threads or the head.
check the limit screws - the chain must NOT be able to jump over the top of the biggest cog.  It should be barely just able to stay there in normal riding, because frame flex and your leg power can give it the chance to get over the top and that will break things.  Perhaps not so bad with your super short spokes, but not good.   The lower limit screw does the same, but at the small cog a chain off can damage your frame and axle.  Neither is good.