Distance from Mysore City to Shravanabelagola is 82 km. By car you can cover this under 2 hours.
The route is like this: Mysore City -- 16km --> Srirangapatna ---1km ---> Junction of SH7 --- 5km ---> Pandavapura Railway station ---32km ---> KR Pet ---14km ---> Kikeri --- 14km ---> Shravanabelagola
Drive out of Mysore city towards Srirangapatna (Bangalore Mysore highway , SH17). After about 16km from the city you will reach Srirangapatna. A little more than a kilometer after Srirangpatna on the highway you will see a left deviation to KR Pet, Channarayapatna etc. There is a signboard just ahead of this junction. This is the start of SH7.
You've left the larger highway and now it is a narrower SH7 that zigzags through many villages. In about 4½km from that deviation you arrives at a railway crossing (Bangalore-Mysore track) gate (that bridge is still under construction!). On the right you'll see the Pandavapura railway station.
Immediately after crossing the gate the road bifurcates. The right fork goes to Pandavapura village, Thonnur Kere, Melkote etc. Take the left fork that leads to KR Pet.
The road condition varies, but it is decent enough for a long drive.
From Pandavapura railway cross to KR Pet is about 32km. Cross this town and head for a village called Kikeri which arrives after 14km from KR Pet. At kikeri ask for the deviation to Shravanabelagola. Turn right into the deviation. It's about 14km drive till you reach the Shravanabelagola town.
It's a bit difficult to find direct buses to Shravanabelagola from Mysore except the Mumbai bound Airavat service (Volvo) from Mysore that goes via Shravanabelagola (1½ hours, Rs134).
An another practical option is to get to Channarayapatna from Mysore. There are many daily express services (Karnataka Sarige) from Mysore that goes via Channarayapatna (2hours, Rs56 ).
From Channarayapatna, look for a local transport ( 14km by car, auto rickshaw, bus etc) to Shravanabelagola town. Another alternative is to head to Hassan and then to Channarayapatna and to Shravanabelagola. During the festival seasons KSRTC operate more frequent services to Shravanabelagola. Check with KSRTC for schedules fares etc.
And of course they operate conducted day tours from Mysore to the Belur, Halebid, Shravanabelagola circuit (see also Mysore to Belur). Contact KSTDC for their packages.