Badami (in Karnataka) is the prime rock climbing spot in India with more than 100 bolted routes (mostly single pitch) and some traditional or top rope routes. Unfortunately the only guide book dates back a few years and is of doubtful quality. It doesn't show all sectors and climbs and the grades are sometimes wild guesses. Nevertheless it is a good start. Probably the better one is to connect with local climbers like Ganesha - they are happy to help. As of now there are 4 main sectors, Temple, Ganesh Plateau, Badami Deluxe and Waterfall (behind the lake). There is a small sector with multi pitch routes not far from Temple but it is accessed from the top.
The rock quality (sandstone) is amazing. The bolting quality varies, some sectors / routes are perfectly bolted, others have rather longer run outs or not well placed bolts and anchors are not always perfect. Bring hangers (10mm, some hangers are missing from easily accessible bolts) and slings to connect anchors!
The challenge in Badami is the sun. Be sure to select a sector in the shade. More sectors or climbs have shade in the morning but you will find routes in the shade at any time of day.
Savandurga is about 60km west from Bengaluru. It is a monolithic granite massive with several multi pitch routes just behind the village. Some of the routes are bolted (8mm) but the pitches have typically long runouts. We climbed Beladingalu and at times it is quite challenging to find the next bolt. Anchors are ok (2 bolts) but the last 1 or 2 pitches (very easy) have no anchors. Some of the rock is brittle, especially at the lower pitches so be careful. It is also adviseable to bring some additional gear (friends and nuts). The way down is an easy walk (take right when arriving on top).
We have heard that there is more climbing around Savandurga and of course in Ramanagaram.