if you want to ride a bike, take delhi. remember that is is havy driving and not the lead back way of driving as in thailand. from delhi to the mountains is nice and good and in delhi it is easy to rent or buy a bike for your trip. from delhi to for instance rishikesh is around 8 hour drive. around here you have several beautiful places; shimla, rishikesh, dehradun and so forth or you can go more up north. sometimes it is difficult to go to srinegar -kashmir-for instance because of the snow in winter and some of the passes are closed then as well.
if you plan to go to mumbay it is better not to try to drive to the himalayas but drive the beautiful roads form kerala, goa and tamil nadu. this trip is better to do from oct. till march. the north better to do from march to sept.