There two major routes. The first is via Kanakapura (that is NH208) and the second route is by the Bangalore to Mysore highway (see Bangalore to Ooty route).
On the Kanakapura road, take the left diversion at Malavalli. In about about 30km from the Malavalli you’ll reach Shivanasamudra Falls. The road is narrow, but driveable through scenic landscapes.
Bangalore to Shivanasamudra Falls via Kanakpura (125km): Bangalore –> 50km –>Kanakapura –>50km –>Malavalli –> 25km –> Shivanasamudra Falls
If you prefer the State highway to Mysore, for any reasons, take the left diversion to Malavalli just as you cross the Maddur town on the SH17 (Bangalore Mysore highway). In about 25km through a less bussy road you’ll reach Malavalli. At Malavalli town , cross the junction and continue on the same road, it’s about 25km to the falls site as mentioned earlier.
Bangalore to Shivanasamudra Falls via Maddur ( distance 134km): Bangalore –> 50km –>Ramanagara –>34km –>Maddur–>25km–>Malavalli –> 25km –> Shivanasamudra Falls
Mysore to Shivanasamudra Falls (distance 73km) : From Mysore you’ve two routes for Shivanasamudra. The first is via T.Narsipur and the second is via Malavalli.
Route 1: Mysore –>25km–>Bannur–>23km–>Malavalli–>25km –>Shivanasamudra Falls
Raoute 2: Mysore –>35km–>T.Narsipur–> 45km –>Shivanasamudra Falls
Also checkout Bangalore to Talakadu and the Cauvery Trail for other travel ideas.