Is there a generic function to find the area of any shape? Given the coordinates of all points making up a two-dimensional shape and the order in which they are connected, is there any generic formula that will give the area of that shape?

EDIT: The list of points would be a list of all of the vertices.
If you mean arbitrary sets of points in the plane, the answer is probably "no" unless you regard the defintion of Lebesgue measure as such a formula.  However, you write about "the order in which they are connected" and that makes me suspect you may have in mind a sequence of points on the boundary with line segments between them as components of the boundary.  In that case, maybe the shoelace formula will help you.

And if you mean polygons whose corners are integer points, there is Pick's theorem.
I believe so, using shoelace theorem. If you line up the co-ordinates starting anywhere but always going counter-clockwise and ending with the first co-ordinate you can multiply the previous x value and after y value and then subtract vice versa and divide by 2. Just search it up....