I think it's good to be grounded in the basic concepts before jumping into specific software. See if your nearest uni library has Mapping by David Greenhood, and possibly his followup Down to Earth: Mapping for Everybody. They're both fun books to read over a weekend, not tedious textbooks.
Next—mostly for fun—use some tracing paper to draw a special-purpose map from a general one. So trace and label something—the route to grandma's house, perhaps—on a metro or provincial road map. You're thinking about what to put in and what to leave out; what to simplify and abstract and displace for readability. Don't spend more than an hour or two on it.
Now you have some idea where you're headed when you make a map, and you're ready to work with some specialized software. You might choose QGIS (open-source) or the ArcGIS home edition ($100). You can get the OSM data from Geofabrik or in other ways, and bring in your waypoints, style it, export to Adobe Illustrator format, then make it look nice using that or Inkscape.
If you know the end product will always be for the web, you can do the styling like all the cool kids do now, using TileMill and Leaflet. Just know that your ability to manipulate or add to those graphics is quite limited, and you'll probably need some kind of GIS tool or programming help (may be as simple as Google Earth and a small utility like KMLCSV from GitHub) to work with your own tracks and waypoints. And what you're creating are raster tiles, which won't print as nicely or as large as vector graphics like you get from Illustrator.
Come back and visit the forum often as you get going. We'll happily encourage you and try to help.