The equation for the tangent plane at a point is:
Find your partial derivatives, plug everything in, and simplify.
To find the normal line to this tangent plane, you then want to define x, y, and z parametrically. They will all have the form:
except with the appropriate variable in place of x. You then solve each equation for t and set all 3 equations equal (since t=t=t, this must be true).