I would like to compare different methods of finding roots of functions in python (like Newton's methods or other simple calc based methods). I don't think I will have too much trouble writing the algorithms
What would be a good way to make the actual comparison? I read up a little bit about Big-O. Would this be the way to go?
The answer from @sarnold is right -- it doesn't make sense to do a Big-Oh analysis.
The principal differences between root finding algorithms are:

rate of convergence (number of iterations)
computational effort per iteration
what is required as input (i.e. do you need to know the first derivative, do you need to set lo/hi limits for bisection, etc.)
what functions it works well on (i.e. works fine on polynomials but fails on functions with poles)
what assumptions does it make about the function (i.e. a continuous first derivative or being analytic, etc)
how simple the method is to implement

I think you will find that each of the methods has some good qualities, some bad qualities, and a set of situations where it is the most appropriate choice.
Comparing Root-finding (of a function) algorithms in Python