How can I please have a trip line a cross the earth between two locations be automatically drawn likewise distance measuring and be able to safe it as part of my session.

The red distance line cannot be saved in a session, because "line measurement" is just a tool mode (just like zoom and pan modes). You can however create a user point map with the two (or more) points (e.g., begin and destination of the trip), and save the map.

Alternatively, you could use any of the GIS desktop tools to create a map with line(s) and labels on your desktop computer, and then convert the map from shapefile to prettymap format using the free online shapefile-to-prettymap converter.