All the best in 2015 to all of you!

Thank you Midgardsormr for the useful link. It helped a lot. I followed the steps and imported the data (gpx reformated to shapefile format) to Tilemill. I used the shapefile road data sets, so there would be no need to use Mapbox for a background (the more can be done localy the better :-)). The only thing I wasn't able to solve, is the "heatmap" coloring (the more is a particular segment used, the more it should be colored - a scaled intensity; or how should I call it).

I came to a conclusion, which can be summarised as follows:
Tilemill is not able (or I wasn't able) to 1) align tracks to a road grid, 2) each track (even if it was done on the same segment of a road) may be displaced in relation to another due to inaccuracy of GPS and GPS recording application, 3) this results in multiple lines being next to each other instead of being on top of the previous. Thus the colored line(s) will not result in a traditional heatmap (compare the cut-outs below).

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Anyway thank you for guiding me in a good direction which resulted in a nice map of my rides (rather sufficient for my level of drawing knowledge).