Bangalore to Belur is about 225 km. Consider 3½ hours driving time. Add and hour more, for what it takes to get out of Bangalore!
Hassan, at a distance of about 40km (25 miles) from Belur is the ideal gateway town for both Belur and Halebid.
Bangalore to Belur by bus: There are a few direct bus services from Bangalore to Belur. Your chances are many fold , if you are prepared to take two bus journeys , that is first Bangalore to Hassan and then Hassan to Belur.
About the direct bus service first. KSRTC operates close to 6 daily services (Karnataka Sarige services) from Bangalore to Belur. Roughly every hour or so from 6 in the morning buses are operated to Belur. Also there is a Rajahamsa service with some more comfortable seats.
Bangalore to Hassan bus options are more with a dozen or so Karnataka Sarige services, 7 Rajahamsa services and an Airavath service. It takes about 4 hours from Bangalore to Hassan.
From Hassan to Belur is another one hour plus journey by connecting bus service. Belur is about 40 km (25 miles) from Hassan. Bus services are operated from Hassan bus station.
Bangalore to Belur by road:
Take NH 48 that goes towards Mangalore. Near Hassan , the Mysore-Hassan state highway crosses NH 48. Take the right deviation towards Hassan town. And then drive towards Belur. Bangalore-Hassan is about 185km (115 miles).
Bangalore to Belur by train:
Hassan is the nearest major railway station for Belur. Trains take a longer route than road to reach Hassan. Train No: 6517/6516 Mangalore Express from Bangalore to Mangalore travel via Hassan. This leaves late night from Bangalore and reaches Hassan early morning. Mangalore Express via Mysore. In any case if you are prepared to travel to Mysore, there is one more daily Express train to Hassan from Mysore ( No.7301 Mysore-Dharwad Express ) and also a couple of weekly express trains.
From Hassan, you've to take the road route to Belur.