How do you find the slope of a line parallel to 2y=6x+18?

If you divide both sides of this equation by $2$ you get:
$y = 3 x + 9$
This is the equation of the line in slope-intercept form, with slope $m = 3$ and intercept $c = 9$.
Any line parallel to it will have the same slope, $3$.
Incidentally, any line perpendicular to a line of slope $m$ will have slope $- \frac{1}{m}$. So in your case, any line perpendicular to your line will have slope $- \frac{1}{3}$.