Answer
vertex: $(0,0)$
Focus: $(0,\frac{3}{2})$
Directrix: $y=-\frac{3}{2}$
See the attached figure.
Work Step by Step
$x^2=6y$
or, $x^2=4(\frac{3}{2})y$
Thus, vertex: $(0,0)$
Focus: $(0,\frac{3}{2})$
Directrix: $y=-\frac{3}{2}$