I would start with find FindFit or NonlinearModel. Those are roughly the same in terms of algorithm, coming to the same results. But NonlinearModel makes an object that carries measure of goodness of fit, the function, the parameters, variance, co-variance numbers, and other such numbers. I've been fairly happy with those functions. Then you can change the degree of the polynomial as you see fit and see how changing the polynomial degree changes the goodness of fit values.
