The Path
This path starts near the Commonwealth Pool. The start is well hidden - to find it turn left just before entering Holyrood Park and then turn right, doubling back (the sign is on the left and isn't very obvious) after a hundred yards or so.
The route starts by going through a long tunnel - a 1 in 30 downhill gradient. It is lit, but not very well, so take care, particularly with pedestrians.
It is also possible to join the path from the park - turn right as you enter and head towards Duddingston. A few hundred yards on there is a turning to the right for cycles and pedestrians. Take care - the road is fairly narrow and often busy - except on Sundays when it is closed to traffic. There is a sharp left just in front of a private drive which then joins up with the main path just after the tunnel described above.
The route continues to Duddingston - where there is a crossing to join up with the next section to Bingham.
Another crossing will take you onto the next section to Brunstane, where there is a choice of continuing or turning left. The signing is a bit confusing - if you're planning on going on to Musselburgh I recommend going straight on and continuing to follow the signs. If you want to go to Joppa and Portobello then turn left and rejoin the road after a few hundred yards, going past the former Big W.
Historical information on the Innocent Railway can be found on Wikipedia.