Finding the Regression of a 3D Scatter Plot

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I have a set of 3D Data, and would like to find an equation that describes the data. The equation will be used to approximate a Z value given an (x,y). I believe that due to the "shape" of the data a non-linear regression will be needed.

I would be very grateful if someone could walk me through this process (I have never taken a stats class). Sorry if this devolves into an "explain like I'm 5" kinda post.