How can I increase the line thicknesses in my exported plots?

You can increase the default thickness values in the Axis Options and Plot Options dialogs. A good starting point would be to set these to 200%. This will prevent the ticks, grids, lines, and markers from becoming very faint or disappearing altogether. The Line Thickness setting in Plot Options will make variable lines, curve fit lines, and error bars thicker. The Marker Thickness field in Plot Options will make the markers thicker.