I believe I know what your problem is.
To move a row, it matters where you do the 2nd click. At least for me it does, and I'm running Numbers 3.1 (which I detest... I want to go back to the previous Numbers with the inspectors).
First, you should highlight the entire row - you do this by clicking the row number at the far left of the table.
Second, after a brief moment, you click-and-hold on the row number AGAIN. While holding, the row will appear to "pop out" and then it can be dragged while you continue to hold.
Releasing the mouse click will set the row in place.
If you don't click (the second time) on the Row number (this is the blue area at far left), it will only attempt to make a new selection. I personally preferred the older method from the previous Numbers, where the mouse cursor changes into a HAND ICON when the row can be moved - this was so much more intuitive.
The new Numbers is extremely frustrating to me and other Numbers veterans. It seems that in an effort to improve, many reliable and well-used features are now made more complicated, and the matter is further compounded by lack of good instructional material.