As the first step please have a look into the contributing guide.
Now which issues to work on?
GraphHopper is over 10 years old and many “easy” problems are likely already solved. Still we try to tag “good first issues” and you can also find a different problem you want to work - probably start discussing them here first to get feedback of possible traps. Sometimes you can contribute just by comparing your daily routes with the routes GraphHopper suggests and create an issue if there are unsafe or illogical suggestions.
It is often also a good entry point to start contributing to GraphHopper Maps (the GraphHopper UI) or to the Navigation SDK to integrate GraphHopper navigation in mobile application. (There is soon also a second option available for navigation.)
One important other issue is the translation: have a look at GraphHopper Maps in your language and see if there are missing, hard to understand or incorrectly translated words visible and fix them 
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GraphHopper needs also your help in our documentation. Especially the API documentation is a bit weak and only the routing and isochrone API is documented. E.g. /navigate, /mvt, /spt and /match are missing. Some documentation is about how you configure it like this blog post about a world wide setup which likely needs to be updated.
Furthermore there is a Java client and JavaScript client, which you can try and see if it is easy to integrate it in your Java or JavaScript projects and help us improve this experience.
Lastly: just spread the word! It helps us if you write about GraphHopper and use it for your next project. It is especially important to improve the first steps with GraphHopper be it the client, the web jar file or the Docker image.