Plotting route as an overlay on a map image

Hi I am not an expert in mapping but am using osrm API along with my own code to calculate routes. I would like to visualise these routes by plotting them on a map - I would like different colours for routes on foot and by car which I have saved in different files (as lists of geocodes).

I realise that for people experienced in the field, this is probably a really simple question, but my internet searches have resulted in me being more confused than before I started. Could anybody help me out?

Thank you all in advance