For web, I’d recommend Leaflet.js. It gives you an easy interface to tile-based map services, and you can write your own JS code to control it. The examples should get you going.
Also take a look at KNReise, it’s a great collection of Open Data demos using data from Norkart and other Norwegian sources. They use Leaflet and the code is available on GitHub.
One tip would be that if you’re doing further calculations (distances, direction etc.) on the basic GPS data, you might consider pre-calculating those values (say, in Processing) and storing them in a CSV. That way you can reduce the amount of computation your code has to do in the browser.
Pre-calculation might not make much of a difference if your data set is relatively small, but if you have >10k data points it might be worth it.