kaputnik noticed this -
British Railways - diagrams of Edinburgh Suburban services, 1958
It's only 50 years since Edinburgh had a fairly extensive suburban rail network. Other lines (e.g. Balerno) had already been shut.
Many have now been converted into routes for other modes of transport.
A lot of the stations (at least the platforms) are still visible, though perhaps most walkers and cyclists don't know their names.
It's not possible to go train spotting, how about some station spotting? Tick them off as you travel through!
http://www.railscot.co.uk is a good source of historic and current railway info.