Use the TinyGPSplus library, that provides very easy to use functions for calculating distance and direction between two sets of co-ordinates, for instance;
The caclulation from TinyGPSplus is accurate to within a metre or so on the checks I have done. I did run a real World test using standard GPS reference points (positions know to sub cm level) to see if the library calculations were good enough.
There are other mathematical methods but I have not checked their real world accuracy, maybe others have.