I need to speed up a function. Should I use cython, ctypes, or something else?

Ignore everyone elses' answer for now. The first thing you should learn to use is the profiler. Python comes with a profile/cProfile; you should learn how to read the results and analyze where the real bottlenecks is. The goal of optimization is three-fold: reduce the time spent on each call, reduce the number of calls to be made, and reduce memory usage to reduce disk thrashing.

The first goal is relatively easy. The profiler will show you the most time-consuming functions and you can go straight to that function to optimize it.

The second and third goal is harder since this means you need to change the algorithm to reduce the need to make so much calls. Find the functions that have high number of calls and try to find ways to reduce the need to call them. Utilize the built-in collections, they're very well optimized.

If you're doing a lot of number and array processing, you should take a look at pandas, Numpy/Scipy, gmpy third party modules; they're well optimised C libraries for processing arrays/tabular data.

Another thing you want to try is PyPy. PyPy can JIT recompile and do much more advanced optimisation than CPython, and it'll work without the need to change your python code. Though well optimised code targeting CPython can look quite different from well optimised code targeting PyPy.

Next to try is Cython. Cython is a slightly different language than Python, in fact Cython is actually best described as C with typed Python-like syntax.

For parts of your code that is in very tight loops that you can no longer optimize using any other ways, you may want to rewrite it as C extension. Python has a very good support for extending with C. In PyPy, the best way to extend PyPy is with cffi.

Cython is the quickest to get the job done, either by writing your algorithm directly in Cython, or by writing it in C and bind it to python with Cython.

My advice: learn Cython.