SHP is several files, limits field names to 10 characters and is not web-friendly. GPX is read by practically every GPS device and fitness app. SHP is at home in exchanging vector map layers between GIS programs; GPX is the usual choice for GPS tracks, routes and waypoints.
Every format is read into plain GeoJSON features and written out from there, so coordinates, names and properties carry over wherever the target has a place for them. A zipped shapefile (the .shp with its .dbf and .shx) or a bare .shp is read. If a .prj file names the coordinate system, coordinates are converted to latitude and longitude; a coordinate system that cannot be read is refused. Points become waypoints and lines become tracks; GPX has no areas, so polygon outlines are written as track segments. Elevation and time are kept. Coordinates outside the latitude and longitude range are refused, since the file is probably in a projected coordinate system.