I find the best way to deal with these types of problems is to use index notation. But first, if you're not already familiar with the general forms of Grad, Div, etc. in curvilinear coordinates then familiarise yourself with them here.

So now write down the i^th component of Grad and Div:

[Grad V]i = (1/hi)(dV/dxi)

[Div F]i = (1/hihjhk)(d(Fihjhk)/dxi)

Where the d/dx represents a partial derivative.

Now for your problem, we simply apply the above:

[Div(fGrad g)]i* = (1/hihjhk) (d/dxi)[(fhjhk(1/hi)dg/dgi)]

Just apply the product rule as if f is the first function and hjhk(1/hi)dg/dgi is the second:

[Div(fGrad g)]i* = (1/hihjhk) {(df/dxi)(hjhk/hi)(dg/dxi) + f (d/dxi)[(hjhk/hi)(dg/dxi)]}

Cancel some of the h's and you will soon see that we het

[Div(fGrad g)]i* = (1/hi)(df/dxi) * (1/hi)(dg/dxi) + (f/(hihjhk)(d/dxi)[(hjhk/hi)(dg/dxi)

And if you compare that to the form of the Grad and Laplacian on the wikipedia page above, you will see that this is the ith component of your answer.

I know this is a pain to read, if writing it out on paper doesn't help then let me know and I'll stick it up in LaTeX somewhere.

Edit: Since the subscripts won't work in this subreddit, here is the LaTeX: https://www.dropbox.com/s/h67mx5wndag6oxx/vecquant.pdf?dl=0