If you come by train there are two Rly. Stations for Mangalore; they are, Mangalore Jn. and Mangalore Central. Get down at the Mangalore Central railway station and take an Auto to the Bus stand ( say SBI Bus stand to the Auto Rickshaw guy - Mangalore has meter system and it should not cost more than 25-30 Bucks).
Locally run Private transport buses are available in a good frequency. A bus leaves Mangalore every 5 minutes to Udupi (5 Kms from Manipal) and every 15-20 minutes to Manipal. However, be warned that these buses overload passengers like sardines in a tin can and are also known for their "edge of the seat" driving, which is not for the fainthearted. The government run Volvo air-conditioned buses are a costlier option and operate every half an hour. However these are much safer and more comfortable.