The route begins almost right at Arashiyama Station on the Hankyu Line. (If you'd rather start from Kyoto proper, you can also pick it up as it passes through the western part of the city, near Katsura Station on the Hankyu line.) Right out of the station, head straight out and a bit to the right and you'll find this sign, showing a small park and the red line of the bikepath leading along the riverbank. Interestingly, the name of the bikepath on this sign (Katsuragawa Cycling Road) doesn't match the name on subsequent signs - usually a sign of battling bureaucracies at work.
At first, you have to take it on faith that this gravel walking path becomes the bikepath - but at this point you can see the white railing of the official paved bikepath beginning in the distance.