Mangalore to Shringeri distance is around 120 km... You can reach in 3 hours car/bus travel. Karkala is the city you will come across while travelling which is 6o kms from mangalore. Mangalore to Karkala, you have two options via moodbidri and Padubidri. I advice Padubidri now though it covers 5 km extra compared to Moodbidri, road is very nice. You may find a toll booth near Mukka/Mulki while travelling via Padubidri. Moodbidri road is narrow and curved. From karkala, Shringeri road is narrow in many places and goes through forest areas. You can visit a few waterfalls on the way and near to Shringeri.