When you select the cell to paste, click only once. This selects the cell and you can use Paste Formula Results. If you click twice instead, it puts you into text editing mode and you will not be able to use Paste Formula Results.
But I'm not sure I totally understood your question. If you double-click and enter text mode, the normal Paste operation seems to do the same as Paste Formula Results. Both put the results of the formula into the cell.
If you want to paste the actual formula into another cell exactly as it appears in the first cell, you have to double click the cell with the formula to bring up the formula bar, select the formula (Cmd A) , Copy (Cmd C), then click on the other cell, type an =, then Paste (Cmd V).
If you want to paste the formula and have any cell references update to match the new cell, you select the cell, Copy, then Paste in the other cell.