If I’m looking from it directly above, the cube can be moved. But when I do move it, with say grab, It will leave behind the bottom of the cube.
probably because your in proportional mode
de select this proporptionel mode O key and it will move ok all in one piece!
I want to select the thing behind it, and move it as well. I want to select two edges.
Do as MCollett says and then box select the edges with B or circle select with C or lasso select with Ctrl+LMB or multiple select by holding Shift