Certain aspects of the Microsoft Word interface are definitely trickier than others. Reorganizing columns in a table is not as simple as it may seem!
If you need to swap the columns in a table in Microsoft Word document, select a column and drag it to a new position. Select the left-most column (hover vertically above the column in the table so that the mouse pointer displays as a black, downward-pointing arrow and left-click to select) and then move it (white arrow) by left-click dragging it to and dropping it at its new position.
Depending on the content of your table you may need to resize your table first – AutoFit » AutoFit to Contents should do nicely.
The trick is in the selection techniques!